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2919 Eclipse Of The SunThe Speed Of Light The Nature Of Light Einstein's Theory Of Gravity And Acceleration The Relativity Of Time Relativity And Four Dimensional Time Ides Of March Observation And Comment FROM THE CIRCLE TO THE SPHERE (ELEMENTARY SELF - EVIDENT SIMPLE ARITHMETIC)Editor In Chief Of Athena Press, Letter Of Recommendation.Latest UpdatesAnalogue Differential Geometry +n- Natural Geometry Of Curvature.The Natural Circle And Its SquareIntroductionSumeria 1,000 BC.The Sumerian Method For Finding The Area Of A CircleArchimedes Proposition 1 - Archimedes Triangle.Three Times The Radius Squared. Four Quadrants. Squaring The Circle. Calculating The Areas Of Rings. |
Twelve Steps From A Cube To Its Sphere.From The Cylinder To Its Cone. The Areas And The Volumes Of Symmetrical Ovals And Ovoids. The Number of Latitudinal and Longitudinal Degrees to the surface of a Sphere.The Area Of One Degree Of A Spheres Surface AreaThe Number of Three-Dimensional Degrees To The Surface Of A Sphere The Irrationality Of π. Circle - Versus - Circumference. Symmetrical Rationality. Alpha And Omega - Infinity And Eternity. The Twelve Square Phenomenon Of The Pythagoras Theorem. Decimal Fraudulent Monetary System The Case Against Decimalisation by Professor A C Aitken (Formerly Professor of Mathematics, University of Edinburgh) email addresses universometry@gmail.com unialphaomega@hotmail.com |

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__Recommended Sources of ***ACTUAL & FACTUAL*** information__**International News Weather & Documentaries Al Jazeera**

See

See

**Al Jazeera**through the years | Al Jazeeera English -

**YouTube**

https://www.youtube.com/watch?v=SZhehnDgjnY

**Video URL**https://youtu.be/SZhehnDgjnY

**Really Graceful (www.reallygraceful) ***EYE OPENING*** videos YouTube*****Highly Recommended***

**Books**

The New Rulers Of The World by John Pilger

The New Rulers Of The World by John Pilger

*********Highly Recommended*****

**A Secret Country by John Pilger**

**FORMATION OF THE UNIVERSE - 2019 ECLIPSE OF THE SUN**

**Stars to Stardust and Ashes – Ashes and Stardust to Stars – Stars to Stardust and Ashes… ad infinitude…**

To begin…

Reference…

YouTube Video ESOcast 173: First Successful Test of Einstein’s General Relativity Near Supermassive Black Hole

Quote: The whole theories about the formation of the Universe - how the Universe is evolving are based upon one philosophical and fundamental ***ASSUMPTION ***, which is that the laws of physics are valid everywhere in the Universe and at any time in the Universe.

Response

An assumption founded upon hubris: The Laws of Mother Nature are the only valid laws which rule the entire natural realm, and all of the physics (physical aspects) relating to the infinite number of ever-changing and evolving states of chaos and order, which go into the makeup of the infinite and eternal Universe.

And for Physicists to assume that the Observations - Assumptions - Laws and the Theories of physics that have been made by them are all correct and infallible, is of an even greater hubris, than that pertaining to their theological (theorist) creationist counterparts.

*Formation of the Universe – 2019 and 1919 Eclipses of the Sun - Einstein’s Theory of*Relativity Etc. (Revised 19 July 2019)To begin…

Reference…

YouTube Video ESOcast 173: First Successful Test of Einstein’s General Relativity Near Supermassive Black Hole

Quote: The whole theories about the formation of the Universe - how the Universe is evolving are based upon one philosophical and fundamental ***ASSUMPTION ***, which is that the laws of physics are valid everywhere in the Universe and at any time in the Universe.

Response

An assumption founded upon hubris: The Laws of Mother Nature are the only valid laws which rule the entire natural realm, and all of the physics (physical aspects) relating to the infinite number of ever-changing and evolving states of chaos and order, which go into the makeup of the infinite and eternal Universe.

And for Physicists to assume that the Observations - Assumptions - Laws and the Theories of physics that have been made by them are all correct and infallible, is of an even greater hubris, than that pertaining to their theological (theorist) creationist counterparts.

**Regardless as to the entire gamut of disingenuous, childish and grandiose delusional theories and theologies of humankind, which abound concerning the creation of the Universe and life?**

It is an ***Absolute Empirically Proven Law*** of ***Mother Nature’s Physics ***

It is an ***Absolute Empirically Proven Law*** of ***Mother Nature’s Physics ***

**THAT**

**Energy ***May Be Converted*** - ***Energy Cannot Be Created*** - ***Energy Cannot Be Destroyed*****

Therefore… Energy is Indestructible

Therefore…Energy is Infinite & Eternal

Therefore…The Universe is Indestructible

Therefore …The Universe is Infinite & Eternal

Therefore The Universe is…

***All That Has Been***All That Is***All That Will Be***

Regardless of all of the childish opinions and beliefs of humankind’s self – promoted, disingenuous, so-called geniuses.

Therefore… Energy is Indestructible

Therefore…Energy is Infinite & Eternal

Therefore…The Universe is Indestructible

Therefore …The Universe is Infinite & Eternal

Therefore The Universe is…

***All That Has Been***All That Is***All That Will Be***

Regardless of all of the childish opinions and beliefs of humankind’s self – promoted, disingenuous, so-called geniuses.

**Energy is ubiquitous throughout the Universe and it is entirely cyclic in nature throughout the Universe…**

Hydrogen plus Oxygen = Water

Water plus Heat = Steam

Steam minus Heat = Water

Water minus Heat = Ice

Water can be converted (split) into hydrogen and oxygen

Hydrogen can be converted (combine) back into water

Answer: You cannot

Simple analogy…

Water plus heat produces Steam – Which is water molecules in Chaos

Water molecules in chaos condense into water – Which is water molecules in Order

Chaos precedes Order – Order precedes Chaos – Chaos precedes Order…ad infinitude

We and all things exist in a state of order, and we and all things deconstruct, or die and decompose.

And then in due course we and all things are reconstructed and our souls reborn, as we and all things continue to grow and evolve.

Within and between…The two extremes of Chaos and Order which go into the makeup’s of the never ending cycles of life, that our immortal souls come to live and know.

Stars to Stardust and Ashes – Ashes and Stardust to Stars – Stars to Stardust and Ashes… ad infinitude…

Copied and pasted from my website homepage

Subjects: 2019 Eclipse of the Sun + Einstein’s Theory of Gravity And Acceleration + The Speed of Light + The Nature of Light + The Relativity Of Time + Relativity And Four-Dimensional Time

JULY 4: UPDATE (Revised [further simplified] 15 & 17 July)

Regarding the eclipse of the sun that has just taken place over Chili, and physicists on-site again claiming that the observation of light bending around the sun, serves to prove Einstein's disingenuous, ridiculous and childish theory of relativity.

Quote...

If The Facts

Albert Einstein

Beginning with Einstein’s foremost claim to fame, which was that he predicted prior to the 1919 eclipse of the sun, that light coming from the background stars behind and further away from the Sun would bend around the sun on its way toward Earth due to the force of gravity.

Following the 1919 eclipse and confirmations given by witnesses who had observed the light coming from background stars bending around the sun on its way toward the Earth, the international press media jumped all over this, and had an international news hay day claiming that not only had Einstein correctly predicted (and some of the media even claiming that he was the first to make such a prediction) that incoming light from background stars would bend around the sun on its way to Earth, also served to prove that his General Theory of Relativity was correct.

Bovine excrement

Had the international press been honest brokers rather than money-grubbing looking for a story journalistic sensationalists

This was first described by Isaac Newton in his work Newtonian Mechanics, and many other physicists

Einstein true to form, as with when he used his friend's notes to cheat and pass a mathematics exam, and obviously copied his wife Mileva's work during their physics exam, plagiarized Isaac Newton’s work relating to classical mechanics and orbital mechanics. And then altered Newton’s equations just enough to make it look as if they were his own work, and to provide a basis for his own disingenuously concocted theory.

A theory which began with the totally unsubstantiated claim, that a photon is *considered to be*

A theory after some six years of receiving ridicule among the general community of physicists and theoretical physicists finally gained acceptance by some aged and obviously going senile

And as such the general population of theoretical physicists then had to accept it, for fear of them being ridiculed for not being able to understand it by those leading physicists who had accepted it, and subsequently other physicists among the general population of physicists.

Hence, a more apt title for Einstein’s Theory of Relativity would be the Emperors Cloak of Physics Theory

ACTUAL AND FACTUAL

The NASA space industry programs use Isaac Newton’s gravitational and orbital calculations to calculate the velocities and the approach angles of spacecraft relative to the gravitational fields of planets, to change the trajectories of their spacecraft, and take advantage of the orbital slingshot effect, which adds to their velocities.

Space Flight: The Application of Orbital Mechanics - YouTube

https://www.youtube.com/watch?v=Am7EwmxBAW8

FACT

As opposed to Isaac Newton’s gravitational and orbital calculations, Einstein's General Theory of Relativity and calculations are not used by NASA, because they are useless when it comes to NASA flight engineers having to work out the orbital mechanics of a space flight.

Hydrogen plus Oxygen = Water

Water plus Heat = Steam

Steam minus Heat = Water

Water minus Heat = Ice

Water can be converted (split) into hydrogen and oxygen

Hydrogen can be converted (combine) back into water

__Rhetorical question__: How do you destroy water?Answer: You cannot

Simple analogy…

Water plus heat produces Steam – Which is water molecules in Chaos

Water molecules in chaos condense into water – Which is water molecules in Order

Chaos precedes Order – Order precedes Chaos – Chaos precedes Order…ad infinitude

We and all things exist in a state of order, and we and all things deconstruct, or die and decompose.

And then in due course we and all things are reconstructed and our souls reborn, as we and all things continue to grow and evolve.

Within and between…The two extremes of Chaos and Order which go into the makeup’s of the never ending cycles of life, that our immortal souls come to live and know.

Stars to Stardust and Ashes – Ashes and Stardust to Stars – Stars to Stardust and Ashes… ad infinitude…

Copied and pasted from my website homepage

Subjects: 2019 Eclipse of the Sun + Einstein’s Theory of Gravity And Acceleration + The Speed of Light + The Nature of Light + The Relativity Of Time + Relativity And Four-Dimensional Time

JULY 4: UPDATE (Revised [further simplified] 15 & 17 July)

Regarding the eclipse of the sun that has just taken place over Chili, and physicists on-site again claiming that the observation of light bending around the sun, serves to prove Einstein's disingenuous, ridiculous and childish theory of relativity.

Quote...

If The Facts

*(Truth's)*Do Not Fit The Theory*(Fiction)*Change The Facts*(Lie through your teeth).*Albert Einstein

Beginning with Einstein’s foremost claim to fame, which was that he predicted prior to the 1919 eclipse of the sun, that light coming from the background stars behind and further away from the Sun would bend around the sun on its way toward Earth due to the force of gravity.

Following the 1919 eclipse and confirmations given by witnesses who had observed the light coming from background stars bending around the sun on its way toward the Earth, the international press media jumped all over this, and had an international news hay day claiming that not only had Einstein correctly predicted (and some of the media even claiming that he was the first to make such a prediction) that incoming light from background stars would bend around the sun on its way to Earth, also served to prove that his General Theory of Relativity was correct.

Bovine excrement

Had the international press been honest brokers rather than money-grubbing looking for a story journalistic sensationalists

*(Motto: Never let the truth get in the way of a good story, e.g. as with today's Murdoch Press Media and Fox Fake News Channel)*and researched the scientific information that was readily available at that time, they would have discovered that far from Einstein predicting or being the first to predict that light bends around the Sun due to gravity.This was first described by Isaac Newton in his work Newtonian Mechanics, and many other physicists

*(Note: Both Einstein and his wife Mileva failed the same exam to become physicists each scoring the same score of 59 out of 100. the pass mark being 60 out of 100. And whereas Mileva was failed obviously for being a woman and of peasant background.*

*Einstein was given a pass, but then could not find work in the arena of physics [due to his failure and a tardy reputation] and had to find work in a patents office.)*had witnessed this phenomenon taking place during thousands of observations of eclipses during the centuries after Newton’s death, hence well prior to the 1919 eclipse.Einstein true to form, as with when he used his friend's notes to cheat and pass a mathematics exam, and obviously copied his wife Mileva's work during their physics exam, plagiarized Isaac Newton’s work relating to classical mechanics and orbital mechanics. And then altered Newton’s equations just enough to make it look as if they were his own work, and to provide a basis for his own disingenuously concocted theory.

A theory which began with the totally unsubstantiated claim, that a photon is *considered to be*

*(by him)*mass-less when at rest, and went on to assert that it is because the empty outer-space surrounding a large body of mass possesses a force of curvature namely gravity, light is forced to bend around large masses such as the Sun,A theory after some six years of receiving ridicule among the general community of physicists and theoretical physicists finally gained acceptance by some aged and obviously going senile

*(or tired of being relentlessly badgered and nagged for six years by this idiot)*leading physicists.And as such the general population of theoretical physicists then had to accept it, for fear of them being ridiculed for not being able to understand it by those leading physicists who had accepted it, and subsequently other physicists among the general population of physicists.

Hence, a more apt title for Einstein’s Theory of Relativity would be the Emperors Cloak of Physics Theory

*(Please refer to the drop-down menu and the two pages concerned with and negating Einstein's theories of Special and General Relativity).*ACTUAL AND FACTUAL

The NASA space industry programs use Isaac Newton’s gravitational and orbital calculations to calculate the velocities and the approach angles of spacecraft relative to the gravitational fields of planets, to change the trajectories of their spacecraft, and take advantage of the orbital slingshot effect, which adds to their velocities.

Space Flight: The Application of Orbital Mechanics - YouTube

https://www.youtube.com/watch?v=Am7EwmxBAW8

FACT

As opposed to Isaac Newton’s gravitational and orbital calculations, Einstein's General Theory of Relativity and calculations are not used by NASA, because they are useless when it comes to NASA flight engineers having to work out the orbital mechanics of a space flight.

**C**

**The reason why light bends around the sun during its transit past the Sun on its way toward Earth, has nothing whatsoever to do with as per Einstein's disingenuous claim, that apparently empty space, which of itself does not possess any form of materialistic structure can bend around or can be bent around anything, any more so than thin air can bend around or be bent around anything.**

What we term as being

it is a fact founded upon observations and bounded in logic and rationality (common sense), that without the presence of a lesser level of energy and matter to serve as a transit medium, not light, not gravitational waves nor anything whatsoever, could travel through outer-space, any more so than birds and projectiles could fly without the presence of the invisible transit medium of thin air, or fish swim in the sea without the presence of the invisibly transparent transit medium of water.

As such, the very fact that light travelling for billions of light-years in a direct straight forward trajectory towards our Sun and the Earth, can be seen to diverge from its direct straight forward trajectory and bend around the Sun, serves to demonstrate that it is light of itself that is being affected by the Suns greater mass, rather than the empty space surrounding the Sun.

Concurs with Isaac Newton's Second Law of Motion

Which states that the rate at which a stationary object from its position of rest, or the extent to which a moving object will deviate from its straight-line path, is dependent on the mass of the object and the force that is applied to it.

Rhetorical Question

When out walking on a calm and still day, could the thin air that surrounds us, of itself, exert a force upon our bodies that could cause them to diverge away from our chosen forward path?

Answer

No, because thin air of itself does not possess a force of direction until due to the Earths weather patterns generated by thermal updrafts and currents, it becomes a force of wind.

Therefore it follows that as empty outer-space which has a far-far less density than that of thin air, as it does not possess any atoms to which an external force if such existed could be applied to.

Cannot of itself - does not possess the ability to apply a force upon the photons of light passing by the Sun; because it does not possess any power of direction or power of force whatsoever.

What the bending of light around the Sun does serve to prove is that photons which Einstein claimed do not possess mass

Do possess mass because they obey exactly the same laws of orbital mechanics as described in Isaac Newton's work, which NASA uses to calculate the orbital mechanics of the masses of their space-craft, relative to the creating a slingshot effect, as their spacecraft passes by and bends around the gravitational fields of the greater masses of the planets.

As Above So Below

Put simply, the reason why both photons and spacecraft are subject to a gravitational slingshot effect when passing by large bodies of mass (planets), is because they do possess mass.

As do all forms of energy and matter, ranging in scale from the most infinitesimal particulates of the microcosmic/quantum universe, up to the largest particulates

Mass =

A density of energy and matter which relative to its size/limits, emits a surrounding gravitational field of influential attraction, that is relative to the density of energy and matter present, relative to size/limits of the mass

Further: Relative to Einstein's premise that photons do not possess mass when they are at rest, sheer nonsense photons are never at rest.

And it is the infinitesimal levels of energy and matter of their mass, which en-mass, serve to fuel the process of photosynthesis “which is the process whereby green plants build up food material during the presence of light; and en-mass serve to produce electricity via photovoltaic conversion in solar panels.

As Above So Below

On a vastly smaller infinitesimal scale as opposed to photons bending around the enormous mass of the Sun, the bending of light towards a greater amount - quantity - density of energy and matter (in this example the energy and matter within glass) can also be observed taking place within a glass prism, whereby the light entering the glass prism refracts (bends) down toward the greater amount - quantity - density of glass at the base of the prism.

Mechanics briefly: When a light photon passes through an atom of glass, the photon is gravitationally attracted towards the greater amount - quantity - density of energy within the nuclei of the atoms nucleus and its trajectory as with the slingshot effect is altered.

As such, as collectively there are a greater number of glass atoms at the base of the glass prism, and therefore collectively, a greater amount - quantity - density of energy at the base of the glass prism, light is refracted down toward the base of the glass prism.

This is the reason why the electromagnetic waves of photons become separated into differing colour bands during refraction, as the photons within each particular colour band have a different amount - quantity - density of electromagnetic energy within their structures. And this is why, taking into account that all of the photons are travelling at the same speed, regardless of the amount - quantity - density within their structures.

It is the photons with the greater amount - quantity - density - of electromagnetic energy within their structures, which are more subject to the gravitational attraction of the nuclei of the atoms.

Because they of themselves have a greater amount - quantity - density of electromagnetic energy within their structures, and therefore they of themselves exert a greater gravitational attraction towards the nuclei of the atoms, than do the photons with a lesser amount - quantity - density of electromagnetic energy within their structures.

Simply: E.g If one type of photon possesses twice the amount - quantity - density of electromagnetic energy as another photon and is travelling at the same speed past the nucleus of an atom, its trajectory will diverge towards the nucleus of the atom by an extra 50% to that of the divergence of the photon with half the amount - quantity - density of electromagnetic energy within its structure.

On a grander and more spectacular scale, we can observe this same phenomenon occurring when we are lucky enough to see a rainbow in the sky.

During this natural phenomenon, we able to observe the full-colour range of the photons of the electromagnetic spectrum, as they are refracted through and from a body or pool of water on the earth's surface up into the sky.

And then as they (as with any projectile of energy within matter e.g. a snowball, a bullet or a missile) lose energy from their upward thrust, they become more subject to the attractive power of the Earth’s gravitational pull below, they curve forward and downward in the form of an arc, as they descend back down towards the Earth's surface.

What we term as being

*(apparently)*empty outer-space is not and cannot be truly empty, it is simply that its fundamental microcosmic energy level falls below that of the methods of the sciences detection levels, as were so-called gravitational waves until quite recently.it is a fact founded upon observations and bounded in logic and rationality (common sense), that without the presence of a lesser level of energy and matter to serve as a transit medium, not light, not gravitational waves nor anything whatsoever, could travel through outer-space, any more so than birds and projectiles could fly without the presence of the invisible transit medium of thin air, or fish swim in the sea without the presence of the invisibly transparent transit medium of water.

As such, the very fact that light travelling for billions of light-years in a direct straight forward trajectory towards our Sun and the Earth, can be seen to diverge from its direct straight forward trajectory and bend around the Sun, serves to demonstrate that it is light of itself that is being affected by the Suns greater mass, rather than the empty space surrounding the Sun.

Concurs with Isaac Newton's Second Law of Motion

Which states that the rate at which a stationary object from its position of rest, or the extent to which a moving object will deviate from its straight-line path, is dependent on the mass of the object and the force that is applied to it.

Rhetorical Question

When out walking on a calm and still day, could the thin air that surrounds us, of itself, exert a force upon our bodies that could cause them to diverge away from our chosen forward path?

Answer

No, because thin air of itself does not possess a force of direction until due to the Earths weather patterns generated by thermal updrafts and currents, it becomes a force of wind.

Therefore it follows that as empty outer-space which has a far-far less density than that of thin air, as it does not possess any atoms to which an external force if such existed could be applied to.

Cannot of itself - does not possess the ability to apply a force upon the photons of light passing by the Sun; because it does not possess any power of direction or power of force whatsoever.

What the bending of light around the Sun does serve to prove is that photons which Einstein claimed do not possess mass

*(When at rest? read on).*

Do possess mass because they obey exactly the same laws of orbital mechanics as described in Isaac Newton's work, which NASA uses to calculate the orbital mechanics of the masses of their space-craft, relative to the creating a slingshot effect, as their spacecraft passes by and bends around the gravitational fields of the greater masses of the planets.

As Above So Below

Put simply, the reason why both photons and spacecraft are subject to a gravitational slingshot effect when passing by large bodies of mass (planets), is because they do possess mass.

As do all forms of energy and matter, ranging in scale from the most infinitesimal particulates of the microcosmic/quantum universe, up to the largest particulates

*(Planets and Stars)*of the macro-cosmic universe.Mass =

A density of energy and matter which relative to its size/limits, emits a surrounding gravitational field of influential attraction, that is relative to the density of energy and matter present, relative to size/limits of the mass

Further: Relative to Einstein's premise that photons do not possess mass when they are at rest, sheer nonsense photons are never at rest.

And it is the infinitesimal levels of energy and matter of their mass, which en-mass, serve to fuel the process of photosynthesis “which is the process whereby green plants build up food material during the presence of light; and en-mass serve to produce electricity via photovoltaic conversion in solar panels.

As Above So Below

On a vastly smaller infinitesimal scale as opposed to photons bending around the enormous mass of the Sun, the bending of light towards a greater amount - quantity - density of energy and matter (in this example the energy and matter within glass) can also be observed taking place within a glass prism, whereby the light entering the glass prism refracts (bends) down toward the greater amount - quantity - density of glass at the base of the prism.

Mechanics briefly: When a light photon passes through an atom of glass, the photon is gravitationally attracted towards the greater amount - quantity - density of energy within the nuclei of the atoms nucleus and its trajectory as with the slingshot effect is altered.

As such, as collectively there are a greater number of glass atoms at the base of the glass prism, and therefore collectively, a greater amount - quantity - density of energy at the base of the glass prism, light is refracted down toward the base of the glass prism.

This is the reason why the electromagnetic waves of photons become separated into differing colour bands during refraction, as the photons within each particular colour band have a different amount - quantity - density of electromagnetic energy within their structures. And this is why, taking into account that all of the photons are travelling at the same speed, regardless of the amount - quantity - density within their structures.

It is the photons with the greater amount - quantity - density - of electromagnetic energy within their structures, which are more subject to the gravitational attraction of the nuclei of the atoms.

Because they of themselves have a greater amount - quantity - density of electromagnetic energy within their structures, and therefore they of themselves exert a greater gravitational attraction towards the nuclei of the atoms, than do the photons with a lesser amount - quantity - density of electromagnetic energy within their structures.

Simply: E.g If one type of photon possesses twice the amount - quantity - density of electromagnetic energy as another photon and is travelling at the same speed past the nucleus of an atom, its trajectory will diverge towards the nucleus of the atom by an extra 50% to that of the divergence of the photon with half the amount - quantity - density of electromagnetic energy within its structure.

On a grander and more spectacular scale, we can observe this same phenomenon occurring when we are lucky enough to see a rainbow in the sky.

During this natural phenomenon, we able to observe the full-colour range of the photons of the electromagnetic spectrum, as they are refracted through and from a body or pool of water on the earth's surface up into the sky.

And then as they (as with any projectile of energy within matter e.g. a snowball, a bullet or a missile) lose energy from their upward thrust, they become more subject to the attractive power of the Earth’s gravitational pull below, they curve forward and downward in the form of an arc, as they descend back down towards the Earth's surface.

****

**THE SPEED OF LIGHT**

****

Index of Refraction

Vacuum

1.0000 <--lowest optical density

Air

1.0003

Water

1.333

Light Flint Glass

1.58

Dense Flint Glass

1.66

Diamond

2.417 >-- Highest optical density

The speed of light when travelling in a straight forward A to B motion of direction through a vacuum here on Earth, within its surrounding gravitational field, has been measured as travelling at a speed of 300 kilometres per second.

Questions

What is the speed of light travelling through interstellar-space when it is not surrounded by a gravitational field, or being subjected to the gravitational influences of any gravitational fields?

What is the refractive index for outer-space and interstellar space?

Fact

Given that the speed of light is variable according to the density of the medium it is being transmitted through, the speed of light is not constant as claimed.

Therefore it is not acceptable for physicists to authoritatively state and mandate, that light does travel universally at a constant speed of 300, 000 km per second when the only measurements they have taken relate to the speed of light as it is here on the Earth.

The Nature of Light

Contrary to the accepted view light is not made up of oscillating electromagnetic fields, nor is it made up of two-dimensional wavelengths which they appear to be when viewed on a flat-screen.

Electromagnetic particles (Photons) travel in a forward three-dimensional spiralling motion, which is why when viewed in a flat two-dimensional cross-section form, on a flat-screen they appear to be travelling in waves.

Empirical Proof of Spiralling Electromagnetic Motion

Acquire a common and readily available plasma ball and view it in a darkened room, and you will observe that the tendrils of electromagnetic light spiral from the Tesla coil sphere, as they travel toward and then radiate outward on the inside of the surrounding glass globe

And it is because of the three-dimensional forward spiralling motion electromagnet particles, that the spirals of electromagnetic emitted from the opposing polarities of permanent magnets are able to exert either a rotational force of repulsion or exert a rotational force of attraction towards another polarity.

Think in terms of two coils intertwining to form a single double coil

And essentially this is the same process of (spiralling) mechanics which is taking place between two permanent magnets but it is as invisible to the naked eye, as are the infrared particles you fire from your remote control toward your TV.

Quote...

For Those Who Believe No Proof Is Necessary, For Those Who Do Not Believe No Proof Is Possible.

Stuart Close

Credibility

Not since the times of the Ancient Sumerians some four millennia ago, and the fall of Babylon to Alexandria of Macedonia in 331 BC...

Has anyone or any group of any of the billions of people who have lived and died over the millennia, or of the seven billion and more people alive today, despite their being given all the benefits of modern computer and IT technology, been able to calculate the exact length to a circles edge.

Given a "

1. Multiply the 120-centimetre Diameter Distance by 3.

2. The length of Distance to the length of the Circle's Circuit is 360-centimetres.

3. The length of Distance to the length of the Circle's Circuit is 360-degrees.

4. Each degree of Distance to the length of the Circle's Circuit is 1-centimetre in length.

5. The length of a circle is three-quarters (75%) that of the perimeter length to its square.

6. The perimeter Distance to the length of a square is four times the Diameter Distance of its inner-circle

And given that all of the elementary - simple arithmetic I have used above and below on this Home Page is self-evidently true, to both academics and laypersons alike.

Every Tutor or scholar who chooses to continue to disingenuously teach, preach and promote the childish decimal - digital decimating system of virtual mathematics, over that of the cohesive imperial analogue mathematics, of the laws of our Cosmic Mother Nature...

Needs to consider that for every action there is an equal and opposite reaction, as we sow so do we reap, what goes around comes around, as above so below...

And regardless of human beliefs or disbelief's, there are and will always be both physical and metaphysical esoteric consequences, relative to any and all of our thoughts and our actions; be they for good or ill.

Index of Refraction

Vacuum

1.0000 <--lowest optical density

Air

1.0003

Water

1.333

Light Flint Glass

1.58

Dense Flint Glass

1.66

Diamond

2.417 >-- Highest optical density

The speed of light when travelling in a straight forward A to B motion of direction through a vacuum here on Earth, within its surrounding gravitational field, has been measured as travelling at a speed of 300 kilometres per second.

Questions

What is the speed of light travelling through interstellar-space when it is not surrounded by a gravitational field, or being subjected to the gravitational influences of any gravitational fields?

What is the refractive index for outer-space and interstellar space?

Fact

Given that the speed of light is variable according to the density of the medium it is being transmitted through, the speed of light is not constant as claimed.

Therefore it is not acceptable for physicists to authoritatively state and mandate, that light does travel universally at a constant speed of 300, 000 km per second when the only measurements they have taken relate to the speed of light as it is here on the Earth.

The Nature of Light

Contrary to the accepted view light is not made up of oscillating electromagnetic fields, nor is it made up of two-dimensional wavelengths which they appear to be when viewed on a flat-screen.

Electromagnetic particles (Photons) travel in a forward three-dimensional spiralling motion, which is why when viewed in a flat two-dimensional cross-section form, on a flat-screen they appear to be travelling in waves.

Empirical Proof of Spiralling Electromagnetic Motion

Acquire a common and readily available plasma ball and view it in a darkened room, and you will observe that the tendrils of electromagnetic light spiral from the Tesla coil sphere, as they travel toward and then radiate outward on the inside of the surrounding glass globe

And it is because of the three-dimensional forward spiralling motion electromagnet particles, that the spirals of electromagnetic emitted from the opposing polarities of permanent magnets are able to exert either a rotational force of repulsion or exert a rotational force of attraction towards another polarity.

Think in terms of two coils intertwining to form a single double coil

*(Helix)*or when you are screwing a screw into a piece of wood. You rotate the screw to the right and the screw is attracted into the medium of the wood, you reverse this and rotate the screw to the left, and the screw retracts out of the medium of the woof and forces it away from itself.And essentially this is the same process of (spiralling) mechanics which is taking place between two permanent magnets but it is as invisible to the naked eye, as are the infrared particles you fire from your remote control toward your TV.

Quote...

For Those Who Believe No Proof Is Necessary, For Those Who Do Not Believe No Proof Is Possible.

Stuart Close

Credibility

Not since the times of the Ancient Sumerians some four millennia ago, and the fall of Babylon to Alexandria of Macedonia in 331 BC...

Has anyone or any group of any of the billions of people who have lived and died over the millennia, or of the seven billion and more people alive today, despite their being given all the benefits of modern computer and IT technology, been able to calculate the exact length to a circles edge.

Given a "

__Diameter Distance__" of 120-centimetres.1. Multiply the 120-centimetre Diameter Distance by 3.

2. The length of Distance to the length of the Circle's Circuit is 360-centimetres.

3. The length of Distance to the length of the Circle's Circuit is 360-degrees.

4. Each degree of Distance to the length of the Circle's Circuit is 1-centimetre in length.

5. The length of a circle is three-quarters (75%) that of the perimeter length to its square.

6. The perimeter Distance to the length of a square is four times the Diameter Distance of its inner-circle

And given that all of the elementary - simple arithmetic I have used above and below on this Home Page is self-evidently true, to both academics and laypersons alike.

Every Tutor or scholar who chooses to continue to disingenuously teach, preach and promote the childish decimal - digital decimating system of virtual mathematics, over that of the cohesive imperial analogue mathematics, of the laws of our Cosmic Mother Nature...

Needs to consider that for every action there is an equal and opposite reaction, as we sow so do we reap, what goes around comes around, as above so below...

And regardless of human beliefs or disbelief's, there are and will always be both physical and metaphysical esoteric consequences, relative to any and all of our thoughts and our actions; be they for good or ill.

**EINSTEINS THEORY OF GRAVITY AND ACCELERATION**

**According to Einstein's theory of gravity and acceleration, he stated that If a lift was taken into outer space and a light beam was shone laterally across the width of the lift, if the body of the lift was then upwardly accelerated the light beam would bend downward, due to the downward acting (pushing) force of gravity.**

**Therefore, given that the**

**speed of the light**

**travelling across the lift is travelling at**

**300, 000 kilometre's per second**

**, it incontrovertibly follows.**

**That in order for the beam of light to bend under a force of acceleration,**

**the amount of acceleration applied to the lift**

**would have to**

**exceed 300. 000 kilometres per second.**

**Therefore, as Einstein stated and maintained all of his life and the rest of world of physicists agreed with him, that**

**it is impossible for anything to travel faster than the**

**s**

**peed of light, it is totally disingenuous to then claim that**

**the mass of a lift can exceed the speed of light.**

**GRAVITY AND NEWTON’S APPLE**

**Oxford English Dictionary**

**Gravity Physics: The force that attracts a body towards the centre of the Earth, or toward any other celestial body.**

**Gravitation: A force of attraction “between any particle of matter” in the universe “and any other”: The effect of this esp, the falling of bodies to earth.**

Disambiguation

Gravity Physics: The force of attraction acting between any particle of matter and any other particle of matter, or acting between any body of matter and any other body of matter, for example;

1. The force of attraction acting between the gravitational centre of the Earth and any incoming celestial body from outer space.

2. The Force of attraction acting between the gravitational centre of the Earth, and any body of matter that is travelling over and above the surface of the Earth, or has been dropped from a height over and above the surface of the Earth.

Disambiguation

Gravity Physics: The force of attraction acting between any particle of matter and any other particle of matter, or acting between any body of matter and any other body of matter, for example;

1. The force of attraction acting between the gravitational centre of the Earth and any incoming celestial body from outer space.

2. The Force of attraction acting between the gravitational centre of the Earth, and any body of matter that is travelling over and above the surface of the Earth, or has been dropped from a height over and above the surface of the Earth.

__Newton's Apple__**Given that one apple is observed as to be falling in the Earth's Northern hemisphere and another apple is observed as to be falling in its Southern hemisphere, neither apple is actually falling.**

1. The apple in the Northern hemisphere is travelling inward from above the surface of the upper Northern polar direction of the Earth, toward the planet's centre of gravitation/gravity.

2. The apple in the Southern hemisphere is travelling inward from above the surface of the lower Southern polar direction of the Earth, toward the planets centre of gravitation/gravity.

1. The apple in the Northern hemisphere is travelling inward from above the surface of the upper Northern polar direction of the Earth, toward the planet's centre of gravitation/gravity.

2. The apple in the Southern hemisphere is travelling inward from above the surface of the lower Southern polar direction of the Earth, toward the planets centre of gravitation/gravity.

**Therefore Newton’s Apple did not fall to the ground in the northern town of Grantham; it was travelling in a southerly direction toward the Earth's centre of gravity, when the ground suddenly got in its way.**

**Quite simply**

**Although it is acceptable in regard to the perceptions of the everyday citizen, for them to think and speak in Euclidean two-dimensional terms as to there being two directions of upward and downward, and that things fall down toward a flat Earth, rather than gravitate toward its curved surface and its centre of gravity.**

**This type of Euclidean linear and flatland thinking, has no place and should be disregarded in relation to the modern day scientific studies of natural geometry, differential geometry, mathematics, astrophysics, electromagnetism, gravitational mechanics, and quantum (micro-cosmic) mechanics.**

**In reality there is no such thing in universal terms as to there being a direction of up or down, north south east or west, or of anything being straight linear in nature.**

**Directions are based upon the fixed landmark and reference points that we impose upon our planet and the surrounding cosmos and use as means of providing fixed navigational reference points, in regard to our travels and our studies.**

Hypothetically (and a little ridiculous) if we could ask Newton's apple which way was up or down, it would have no idea, and we would have to tell it which way was up and which way was down.

Hypothetically (and a little ridiculous) if we could ask Newton's apple which way was up or down, it would have no idea, and we would have to tell it which way was up and which way was down.

**THE RELATIVITY OF TIME**

**Imagine that you are dreaming and in your dream, you are back in your mothers’ womb safe warm snug and simply floating in space, and as you look around you all you can see is diffuse light.**

**Where are you?**

**You are nowhere, as there is no one to tell you where you are and there are no landmarks to relate to or places to be.**

Who or what you?

You are no one and no-thing, as there is no one or any other thing for you to be relative to, and you have no knowledge as to what you are or what you may be.

Who or what you?

You are no one and no-thing, as there is no one or any other thing for you to be relative to, and you have no knowledge as to what you are or what you may be.

**What time is it?**

**You are no when, as there are no sundials clocks or calendars for you to relate to, and as such your existence is entirely timeless.**

**All you know is that you exist because you think and therefore you are, but time most certainly does not**

*.*****

*RELATIVITY AND FOUR - DIMENSIONAL TIME*

**HYPOTHETICAL**

****

**Most of us have seen or handled a golf ball at some time and observed that it has small equally spaced indents into its surface, that give the golf balls surface an all over dimpled appearance.**

**Keeping this image in mind, let's imagine that instead of being just a golf ball it is similar to the Death Star which appears in Star Wars. And rather than it being small and white it is a huge silver sphere out somewhere in space with a diameter measuring ten kilometres across.**

**The number of indents into its dimpled surface is 6000, and each indent is a rocket tube opening with each tube holding of a single rocket manned by a pilot on quick ready alert standby.**

**Given an enemy attack, each rocket is capable of leaving its tube at almost the speed of light, and achieves the speed of light, within one second of leaving their rocket tube.**

**Also, given that there are no directions in outer space, we need to decide which is the top and bottom of the Death Star and so nominate which is North and which is South.**

**Now let's further imagine that we are the Commander in charge of the Death Star, and to hand we have a red master control trigger which is linked to each of the 6000 rocket tubes, on our command and on releasing the control trigger the 6000 engines of the rockets can simultaneously ignite, and carry their pilots into all quadrants of space.**

**Suddenly approaching enemy ships are detected at a distance of 6 light seconds from the Death Star incoming from all sectors of space, and so we release all six thousand pilots and rockets into surrounding space to meet the threat.**

**Rhetorical Question**

Given that 6000 pilots and their rockets left their tubes and achieved the speed of light one second after having they exited their tubes.

Would it be fair to say, given that every one of 6000 clocks onboard the rockets would slow down near the speed of light and cease to tick at the speed of light?

That this would apply to every one of the hearts (tickers) belonging to those 6000 rockets, would also slow down and cease to beat at the speed of light?

Because if so then every one of 6, 000 pilots would be dead before entering into battle.

Given that 6000 pilots and their rockets left their tubes and achieved the speed of light one second after having they exited their tubes.

Would it be fair to say, given that every one of 6000 clocks onboard the rockets would slow down near the speed of light and cease to tick at the speed of light?

That this would apply to every one of the hearts (tickers) belonging to those 6000 rockets, would also slow down and cease to beat at the speed of light?

Because if so then every one of 6, 000 pilots would be dead before entering into battle.

**IDES OF MARCH OBSERVATION AND COMMENT**

**As Posted On YouTube**

**Ide's of March Observation and Comment**

**Regarding the page titled *Jesus Mary And Judas* on my web site.**

**It has been pleasing to note, that since exposing the deceits and the lies within the four anonymously written Roman Gospels of Mathew Mark Luke and John, and posting them on YouTube.**

**I have not received any comments seeking to mitigate or to give support to the lies I have exposed, but rather an overall blanket silence.**

**Which serves to inform me, that despite two-thousand years of Roman Empire Catholic Church invasions, genocides**

*(e.g. of the peaceful Cather’s)*

**, torture**

*(e.g. of the Spanish inquisitions*

**) burning alive at the stake**

*(e.g. of the Knights Templar. Joan of Arc, thousands of women tortured and then drowned, or tortured and then burned alive for supposedly being witches)*

**enslavement's, lies, propaganda, suppression's, and militaristic oppression's.**

**The self-evident truths I have revealed concerning Jesus Mary and Judas, as to what really took place prior to the crucifixion of Jesus two-thousand years ago, and as opposed to the biblical lies and two-thousand years of nefarious self-serving propaganda by the Roman Catholic Church**

**Will hence ensure, that many more questions will be raised concerning the unholy Roman Fascist (**

*Zionist - Privatising - Thieving - Capitalist**)*

**Church's primitively inhumane and vile history, of universally, subversively and insidiously promoting and radicalising violence, misogyny, pedophilia, and depraved orgiastic sexual perversions.**

**The answers to which will lead on toward the fulfillment of the End-time of the two-thousand year’s reign of Evil Empire.**

**The Three Pillars of Zionist Roman Christian Fascism**

**Jewish - Greek - Latin**

**Abraham: Who claimed that God told him that the Jews were his chosen children.**

**Aristotle:**

*(From which the word Aristocrat is derived)*

**who claimed and taught Alexander of Macedonia, that the Greeks were Gods among men, and all other races inferior and should be slaves.**

**Roman: Pirates of the Mediterranean who used a village called Rome as a their base, as they expanded their numbers and their privateering**

*(Privatisation...Invasion/Looting/Rape/Mass-Murder/Oppression/Enslavement)*and pirating operations, and in the process of so doing, wiped out the entire Etruscan race of peoples*(of what is now named Italy)*so that they could take their lands for themselves*(The history of Capitalism and modern day Israel).*

**And working in alliance and on behalf of the deceitful Greek merchants**

*(Beware Greeks bearing gifts)*

**of the Mediterranean, they went on to wipe out all of the Carthaginians, who being fair traders were commercially more successful than the Greeks and hence their major competitors.**

**And as their reward for the bloody massacre and genocide of the Carthaginians, the Greeks gave these murderous pirates and cold blooded mercenary low-life scum, a fake legitimacy as a people, by concocting the (infantile) Roman mythology of Romulus and Remus**

*(A plagiarism of the Cain and Abel story)*

**and a fictional lineage of Roman Kings.**

**Liars criminals and mass murderers then, and over the last two-millennia up to the present day nothing has changed, as they have prospered from the invasions and wars they have instigated while hiding behind the religious facade/cloak of their Fascist Papal Wolf disguised in Glorified Shepherds Clothing.**

**FROM THE CIRCLE TO THE SPHERE**

*(ELEMENTARY SELF - EVIDENT SIMPLE ARITHMETIC*

**Copy of the letter from Mark Sykes the Editor in Chief of Athena Press Publishers 2005, concerning my previous work Universal Geometry - Trigonometry.**

**Quote: Rather to my surprise she seemed to take violently against the book, and although we had a long and quite amicable discussion, I was quite unable to discover her reasons. She praised the clarity and the logic of the presentation and argument but seemed to take against the conclusions. I can only assume that this is because the book challenges received orthodoxy.**

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

**Stuart Close**

**It was not surprising to me that a theoretical mathematician would take violently against my work, because it is not simply a matter of my work challenging the received orthodox teachings relating to mathematics and differential geometry.**

**But rather as she would have known, its publishing and its distribution worldwide would prove to the world on an international scale, that all of the mathematics and theoretical mathematics which have been used for more than two-thousand years relating to the use of pi and the decimal system.**

Are no more than close approximations to that of reality, and therefore of no greater mathematical worth than that of a near guess.

Are no more than close approximations to that of reality, and therefore of no greater mathematical worth than that of a near guess.

**And given the correct and very simple method for calculating the exact length of a circle...**

**Given a "**

__Diameter Distance__" of 120-centimetres.

**1. Multiply the 120-centimetre Diameter Distance by 3.**

**2. The length of Distance to the length of the Circle's Circuit is 360-**

**centimetres.**

**3. The length of Distance to the length of the Circle's Circuit, is 360-degrees.**

**4. Each degree**of Distance

**to the length of the Circle's Circuit, is 1-centimetre in length.**

**It follows that Multi Millions of mathematical theories and theories of physicists such as/e.g. those of an idiot named Einstein must fall, and the trillions of decimal (Ten numeral unit) and digital (1 & 0 = + & -) Euclidean based two-dimensional equations of virtual reality, be converted into the (Three numeral unit) analogue (positive neutral and negative + n -) based three-dimensional (Lateral Vertical Diagonal) and the non-dimensional infinite and eternal cyclic-differential state, of physical and metaphysical-spiritual energy.**

**ANALOGUE DIFFERENTIAL GEOMETRY**

+ n -

The Natural Geometry Of Circles And Curvature

+ n -

The Natural Geometry Of Circles And Curvature

**SIMPLIFIED FOR**

**EVERYONE**

**PARENTS - TEACHERS - ARTS - CRAFTS - TRADES - CONSTRUCTION - TECHNICAL - PROFESSIONS - SCIENCES****THE NATURAL CIRCLE AND ITS SQUARE**

**PREAMBLE**

An example of the natural circle as opposed to a circumferential

The Moon is a solid spheroid that exists within and is in direct contact with the surrounding volume of its outer-space.

As such, and although we see the moon as a large yellow circle outlined against the dark background of the night, the round shape of the moon does not have an outline, because it is simply a solid body that is silhouetted by colour and contrast against the darkness of the night.

When we look at another person or any other solid shape, what we are seeing is not the outline of the person or the solid shape, but rather the shades colours and the textures that go into the makeup of the solid shape, which are silhouetted by contrast against the shades colours and the textures of the background surrounding the solid shape.

An example of the natural circle as opposed to a circumferential

*(outer-lined)*circle, is a full moon being viewed against the dark background of a night.The Moon is a solid spheroid that exists within and is in direct contact with the surrounding volume of its outer-space.

As such, and although we see the moon as a large yellow circle outlined against the dark background of the night, the round shape of the moon does not have an outline, because it is simply a solid body that is silhouetted by colour and contrast against the darkness of the night.

When we look at another person or any other solid shape, what we are seeing is not the outline of the person or the solid shape, but rather the shades colours and the textures that go into the makeup of the solid shape, which are silhouetted by contrast against the shades colours and the textures of the background surrounding the solid shape.

**For example, hold out your hand in front of you and take a good look at it, and consider it in view of our modern day knowledge of its molecular structure.**

**The atoms of the skin of your hand are in direct contact with and merge with the atoms of the gases of the air surrounding your hand, as such no form of lines actually exist to define the shape of your hand.**

**The visual presence of all things, is defined by the mass the size the shape and the colours and textures of things, not the lines of Euclidean linear mimicry.**

__The Natural Circle__**Given a "**

__Diameter Distance__" of 120-centimetres.

**1. Multiply the 120-centimetre Diameter Distance by 3.**

**2. The length of Distance to the length of the Circle's Circuit is 360-**

**centimetres.**

**3. The length of Distance to the length of the Circle's Circuit, is 360-degrees.**

**4. Each degree**

**of Distance to the length of the Circle's Circuit, is 1-centimetre in length.**

__SQUARING THE CIRCLE__**5. Multiply the 120-centimetre Diameter Distance by 4, the Perimeter Length of the Circles Square is 480-centimetres.**

6. The Circle is both 360-centimetres & 360 Degrees in length, which is three-quarters of the length to the Circles 480-centimetres perimeter square.

6. The Circle is both 360-centimetres & 360 Degrees in length, which is three-quarters of the length to the Circles 480-centimetres perimeter square.

__Simply__**Three times the length of…A Line…is the length of the lines Circle.**

**Four times the length of… A Line…is the length of the lines Square.**

__THREE TIMES THE RADIUS SQUARED__**Using a 120-centimeter diameter**

Using the radius

Multiply the 3,600 square centimeters square of the radius by 3, this will yield the sum of 10,800 square centimeters to the area of the circle, which is three-quarters of the 14,800 square centimeters of the square of the circle’s diameter.

*(Diameter Distance)*multiply the diameter by 120, this will yield the sum of 14, 400 square centimeters to the square of the diameter.Using the radius

*(Radius Distance)*of the diameter of 60-centimeters multiply the radius by 60, this will yield the sum of 3,600 square centimeters to the square of the radius.Multiply the 3,600 square centimeters square of the radius by 3, this will yield the sum of 10,800 square centimeters to the area of the circle, which is three-quarters of the 14,800 square centimeters of the square of the circle’s diameter.

__Reader Self-Evidence__**Having read the simple arithmetic above, can you disprove or discredit the simple arithmetic I have used, or convince yourself that a circle is not exactly three times its diameter length?**

For the average

__Rhetorical Questions__For the average

**person and trades-persons.**

Given the choice between having to use Pi

Given the choice between having to use Pi

**3.14285714285 to calculate the approximate length of a circle, as opposed to using 3 times the circles diameter length to find the correct length of a circle, which would you rather use?**

**For Educators and**

**professions.**

**Given the number of tedious and lengthy Pi**

**3.14285714285 approximations**

**carried out each day around the world among a population of more than seven billion people.**

**H**

**ow many trillions of irreplaceable life-hours/days could be saved and be put to better use over the space of a year?**

**INTRODUCTION**

**On reading the web-site content list on the drop-down menu, you might think "whoa this stuff is way over my head" I assure you it is not.**

**If you**

*(and obviously as you are reading this page , you do)*

**have the four basic skills of being able to add, subtract, multiply, and to divide numbers, then none of the elementary arithmetic that follows, will be beyond your mathematical abilities or comprehension, and as such will prove to be self evidently true.**

**To begin, and for the sake of credibility regarding the subject matter, we should first refer to some recent discoveries made in the Fields of Archaeology.**

__Reference__

**: The Guardian Aug 24, 2017: Sumerian Trigonometry Tablet Discovery **

Mathematical secrets of ancient tablet unlocked after nearly a century ...

**https://www.theguardian.com › Science › Mathematics**

__Reference__

**:**Ancient Babylonians Used Geometry To Track Jupiter Thousands Of ...

**www.iflscience.com/space/babylonian-astronomers-used-geometry-study-sky/**

**It is historical fact, as this and other discoveries made in the fields of archaeology continue to confirm, that over a period extending back in time to more than four thousand years ago, that it was the ancient civilization of Sumeria who were the progenitors of: The first alphabet, of writing, of mathematics, of geometry, of differential geometry, of architecture, of engineering, of astrophysics, of clocks, and 3,600 seconds per hour, to the 360-degree twenty-four hour day.**

**And far toward the opposite extreme as to the amoral**

**narcissistic and hedonistic G**

**reeks having been the progenitors of civilisation, democracy and just about everything else**

*(that could not be nailed down)*as well.**It was the intellectually challenged Greek armies of the barbarian Alexander of Macedonia, and their later allies the Romans, who were responsible for the destruction and the loss of thousands of years of knowledge, and of delivering the greatest blow to civilisation and human progress ever known.**

**To the point, as the newly discovered clay tablets of Sumeria serve to prove, and despite all of the disingenuous tripe and**

**propaganda,**

**which the West**

**ern Grecian-Roman**

*(Capitalistic - Fascistic)*U**niversities have continued to dish up over the last two-millennia.**

**Euclid was not the father of geometry, and Archimedes was not the eureka genius he has been made out to be.**

**And the exemplar proof of this is, that**

**despite my not being a Grecian-Roman University (Old-Boy Approved) Euclidean taught geometer or mathematician, and as such a victim of their disingenuous teachings.**

Unlike they and all of their students for more than two-thousand years, who have barely been able to approximate the length to a simple circle.

Unlike they and all of their students for more than two-thousand years, who have barely been able to approximate the length to a simple circle.

**I have here on this homepage, by use of no more than**

__self-evident - irrefutable - simple arithmetic__w

**hich all of you can understand, I have published e.g.**

**A. Four methods for finding the exact area of a circle.**

B. The method for finding the exact areas of rings.

B. The method for finding the exact areas of rings.

**C. The twelve-step method for calculating the exact surface areas and volumes of spheres and ovoids.**

**D. The Number of two-dimensional degrees to the surface area of a sphere.**

**E. The number of three-dimensional degrees to the surface area of a sphere.**

**And as such it should also be noted: That given that the simple arithmetic I have used is self evidently true and**

**irrefutable**

**, so it logically and rationally follows that a**

**ll of this work is solely my intellectual property, and subject to my copyright, and any dispensations I may choose to make regarding that copyright.**

**Reference: How many humans have lived in the past 2013 years? - Quora**

**https://www.quora.com/How-many-humans-have-lived-in-the-past-2013-years**

**And thanks to Alexander of Macedonia, Euclid and Archimedes, to this very day, not one of the millions of so-called mathematicians and g**

**eometer's worldwide, is able to**

**carry out the two simple sums that any Sumerian child could have done four-thousand years ago, which are 3 x a straight line is a circle, 4 x a straight line is a square.**

**SUMERIA 1,000 BC**

**Reference**

**Ancient Babylonians Used Geometry To Track Jupiter Thousands Of ...www.iflscience.com/space/babylonian-astronomers-used-geometry-study-sky/**

**An outstanding discovery could change how we view the history of science. New research has suggested that ancient Babylonian astronomers used geometry to track the position of Jupiter in the sky, one and a half millennia before European thinkers developed the same approach.**

Astroarchaeologist

Dr. Mathieu Ossendrijver of Humboldt University discovered five cuneiform tablets (a type of ancient writing) with detailed calculations that predict how Jupiter would move across the sky. The tablets make reference to the trapezoid procedure, similar to what modern-day physics students use when calculating positions in a velocity-time graph. His findings were published in this week's issue of Science.

“What is new about these tablets is that they mention geometrical figures,

” Dr. Ossendrijver told IFLScience. “Two were already known in the 1950s but all of them are damaged, so they could not be read completely and it was not completely clear that they deal with Jupiter.”

There are over 450 tablets in the Babylonian astronomy corpus. Most of them detail the motion of the Moon and the Sun and are based on the Zodiac, which was invented in Babylonia around the 5th century B.C.E. The tablets in the study are the few describing Jupiter that were associated with Marduk, the patron god of Babylon.

“This is now totally clear because of a 5th tablet,” said Dr. Ossendrijver. “It is the key for the other tablets. What is described on this tablet is the velocity of Jupiter expressed in degrees per day.”

The tablet contains the values of Jupiter’s daily displacement, connecting the trapezoid procedure to real astronomical data. The Babylonians knew that the apparent velocity of Jupiter in the sky is not constant, and they were able to make predictions using abstract geometry. The tablet gives a complete description of the velocity of Jupiter for more than a year.

“The Babylonians and also the Greeks observed that the planets don’t move at a constant speed; sometimes they slow down, they come to a standstill, they go backwards, they come to a standstill, and they move forward again,” added Dr. Ossendrijver. “They do a loop. The Babylonians observed it, described it, and modelled it in mathematical ways.”

In the 14th century, the same procedure was then redeveloped in Oxford and Paris, and it is at the very core of the calculus that was developed by Newton and Leibniz in the 17th century.

Astroarchaeologist

Dr. Mathieu Ossendrijver of Humboldt University discovered five cuneiform tablets (a type of ancient writing) with detailed calculations that predict how Jupiter would move across the sky. The tablets make reference to the trapezoid procedure, similar to what modern-day physics students use when calculating positions in a velocity-time graph. His findings were published in this week's issue of Science.

“What is new about these tablets is that they mention geometrical figures,

” Dr. Ossendrijver told IFLScience. “Two were already known in the 1950s but all of them are damaged, so they could not be read completely and it was not completely clear that they deal with Jupiter.”

There are over 450 tablets in the Babylonian astronomy corpus. Most of them detail the motion of the Moon and the Sun and are based on the Zodiac, which was invented in Babylonia around the 5th century B.C.E. The tablets in the study are the few describing Jupiter that were associated with Marduk, the patron god of Babylon.

“This is now totally clear because of a 5th tablet,” said Dr. Ossendrijver. “It is the key for the other tablets. What is described on this tablet is the velocity of Jupiter expressed in degrees per day.”

The tablet contains the values of Jupiter’s daily displacement, connecting the trapezoid procedure to real astronomical data. The Babylonians knew that the apparent velocity of Jupiter in the sky is not constant, and they were able to make predictions using abstract geometry. The tablet gives a complete description of the velocity of Jupiter for more than a year.

“The Babylonians and also the Greeks observed that the planets don’t move at a constant speed; sometimes they slow down, they come to a standstill, they go backwards, they come to a standstill, and they move forward again,” added Dr. Ossendrijver. “They do a loop. The Babylonians observed it, described it, and modelled it in mathematical ways.”

In the 14th century, the same procedure was then redeveloped in Oxford and Paris, and it is at the very core of the calculus that was developed by Newton and Leibniz in the 17th century.

**.............**

Mathematical secrets of ancient tablet unlocked after nearly a century ...https://www.theguardian.com › Science › Mathematics

Aug 24, 2017 - Dating from 1,000 years before Pythagoras’s theorem, the Babylonian clay tablet is a trigonometric table more accurate than any today, say researchers. ... The 3,700-year-old broken clay tablet survives in the collections of Columbia University, and scientists now believe they have...

.............

__Reference__: The Guardian Aug 24, 2017: Sumerian Trigonometry Tablet Discovery Mathematical secrets of ancient tablet unlocked after nearly a century ...https://www.theguardian.com › Science › Mathematics

Aug 24, 2017 - Dating from 1,000 years before Pythagoras’s theorem, the Babylonian clay tablet is a trigonometric table more accurate than any today, say researchers. ... The 3,700-year-old broken clay tablet survives in the collections of Columbia University, and scientists now believe they have...

.............

__Reference__: Estimating the wealth; Encyclopedia Britannica.

A Babylonian cuneiform tablet written some 3,000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value, which would be correct if π is estimated as 3.

Quote: *which would be correct if π is estimated as 3*

Truth: The Grecian estimate

A Babylonian cuneiform tablet written some 3,000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value, which would be correct if π is estimated as 3.

Quote: *which would be correct if π is estimated as 3*

Truth: The Grecian estimate

**π is a near approximation as to the length of a circle, the correct length of a circle is 3 times its diameter length.**

****

Author: Regarding the comment *which would be correct if

Author: Regarding the comment *which would be correct if

**π is estimated as 3***

**,**

**unbelievable?**

**the sheer lack of generational and personal insight intrinsic to this comment, denotes a degree of stupidity and a level of attendant hubris that is truly astounding.**

**It was the Sumerians who**given their comprehensive knowledge of astronomy, differential geometry, and mathematics,

**were the first to recognize that all circles are identical, perfectly symmetrical, and all are *transitionally proportional* by ratio to all other circles of the Cosmos, regardless as to their size and whether or not they exist in the realm of the physical world, or the realm of the metaphysical mind.**

**All circles have a universally exact *transitional length* that can be mathematically subdivided by any *whole natural number* into any number of equal lengths, or by any number of whole natural numbers which have been holistically subdivided into an equal number of smaller lengths (e.g. 360 degrees of 60 minutes = 3,600 seconds).**

**THE SUMERIAN METHOD**

FOR FINDING THE AREA OF A CIRCLE 1000 BC

FOR FINDING THE AREA OF A CIRCLE 1000 BC

**1U**

**sing a 120-centimeters "Diametric Distance"**

**Multiply, the 120-centimeters by 3**

**The Circle is 360-centimeters long**

Multiplying the 360 centimeters by 360, yields 129, 600 square-centimeters

Dividing the 129, 600 square-centimeters by 12, yields 10, 800 square-centimeters to the circles area

Answer 10, 800 Square Centimetres

Multiplying the 360 centimeters by 360, yields 129, 600 square-centimeters

Dividing the 129, 600 square-centimeters by 12, yields 10, 800 square-centimeters to the circles area

Answer 10, 800 Square Centimetres

**ARCHIMEDES**

**287 - 212 BC**

**Archimedes Triangle**

**Proposition 1.**

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

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

**The circle for which we have to find the area using Archimedes proposition 1, has a 120-centimeter diameter length**

**The base-line length of the triangle is 60-centimeters (which is the radius length of the circle)**

**The right-angle height of the triangle is 360-centimeters (which is the length of the circles 360 degree edge)**

**The 360-centimeter height of the right-angle, is equal to 6 x the radius length**

**Radius length of 60-centimeters x the 360-centimeters to the circles edge yields 21, 600 square-centimeters which is the square area of the rectangle**

**Half of the 21, 600 square-centimeters yields**

**10, 800 square-centimeters, to the square area of the circle.**

**Answer 10, 800 Square Centimetres**

**THREE TIMES THE RADIUS SQUARED**

**

**Pi**

**Circumference Calculator - Omnihttps://www.omnicalculator.com/math/circumference**

**Radius 60 centimetres**

Diameter 120 centimetres

Circumference 376.991

Area 11, 309.73

Diameter 120 centimetres

Circumference 376.991

Area 11, 309.73

**Pi x r2 11, 309.73**

**3 x r2**

__10, 800__**509.73 Square Centimetres of error**

**2017 AD**

**Using a 120-centimetre length "Diameter Distance"**

The diameter x 120- centimetres yields 14, 400 square-centimetres to the square of the diameter

The 60-centimetre radius x 60-centimetres yields 3, 600 square-centimetres to the square of the radius

The square of the radius x 3 yields 10, 800 square-centimetres to the area of the circle, and the area of the circle is 3/4 of its square.

The diameter x 120- centimetres yields 14, 400 square-centimetres to the square of the diameter

The 60-centimetre radius x 60-centimetres yields 3, 600 square-centimetres to the square of the radius

The square of the radius x 3 yields 10, 800 square-centimetres to the area of the circle, and the area of the circle is 3/4 of its square.

**Answer 10, 800 Square Centimetres**

**FOUR QUADRANTS**

******2017 AD**

**Using a 120-centimetre “Diametric Distance”**

**Diameter x 120 yields an area of 14, 400 square-centimetres.**

**Divided by 4 each quadrant of the square will be 3, 600 square-centimetres**

**Divided by 4 each quadrant of a quadrant will be 900 square-centimetres**

**900 square centimetres multiplied by 3 yields 2,700 square-centimetres**

**2.700 square centimetre multiplied by 4 yields 10, 800 square centimetres to the circles area which is ¾ of its square**

** Answer 10, 800 Square Centimetres**

**SUMERIAN METHOD 1000 BC; 10,800 square-centimeters to the circle**

ARCHIMEDES TRIANGLE 212 BC; 10,800 square-centimeters to the circle

THREE TIMES THE RADIUS SQUARED 2017 AD; 10,800 square-centimeters to the circle

FOUR QUADRANTS; 10, 800 square-centimeters to the circle

ARCHIMEDES TRIANGLE 212 BC; 10,800 square-centimeters to the circle

THREE TIMES THE RADIUS SQUARED 2017 AD; 10,800 square-centimeters to the circle

FOUR QUADRANTS; 10, 800 square-centimeters to the circle

****

__FOUR:__SELF-EVIDENTLY Correct and identical results can not be a coincidence.****

**ABSOLUTES****The length to the edge of a circle is 6 times the length of the circle's radius.**

**The length to the edge of a circle, is three-quarters of the length to the square of the circle's diametric distance line.**

**The area to the shape a circle, is three-quarters of the area to the square of the circle's diametric distance.**

**SQUARING THE CIRCLE**

**The area of the first two rectangular progressions of 5/12ths and 4/12ths added together, forms a rectangle measuring 9 squares by 12 squares, thus, containing 108 squares, the same amount of area as the yellow circle in the first diagram.**

**Therefore, given one right angle of the same height as that of a circle - by extension, a rectangle can be drawn that contains the same area as that of a circle of the same height.**

CALCULATING THE AREAS OF RINGS

CALCULATING THE AREAS OF RINGS

**EIGHT MILE DIAMETER RIGHT ANGLED SQUARE**

*(Or Kilometer's)***We begin by first finding the area of each circle**

**Multiply the 2-mile diameter of the central yellow circle by itself = 4 square miles to the square of the diameter, divide by 4 = 1 square mile x 3 = 3 square miles to the central circle.**

**Multiply the 4-mile diameter of the red circle by itself = 16 square miles to the square of the diameter, divide by 4 = 4 square miles x 3 = 12 square miles to the red circle.**

**Multiply the 6-mile diameter of the blue circle by itself = 36 square miles to the square of the diameter, divide by 4 = 9 square miles x 3 = 27 square miles to the blue circle.**

**Multiply the 8-mile diameter of the green circle by itself = 64 square miles to the square of the diameter, divide by 4 = 16 square miles x 3 = 48 square miles to the green circle.**

**Deduct the 3 square mile area of the central yellow circle from the 12 square mile area of the red circle; = 9 square miles to the area of the red ring.**

**Deduct the 12 square mile area of the red circle from the 27 square mile area of the blue circle = 15 square miles to the area of the blue ring.**

**Deduct the 27 square mile area of the blue circle from the 48 square mile area of the green circle = 21 square miles to the area of the green ring.**

**Deducting the 48 square mile area of the green circle from the 64 square mile area of the overall pale blue square, = 16 square miles to the remaining area of the square, which is 1/4 of the area of the square of 64 square miles.**

**Check**

**Central Circle = 3 square miles**

**Red Ring = 9 square miles**

**Blue Ring = 15 square miles**

**Green Ring = 21 square miles**

**Pale Blue Area = 16 square miles**

**Total = 64 square miles**

**FROM THE CUBE TO ITS SPHERE**

**From the Cube to its Cylinder**

**Diagrams 1 - 5 depict the potential cylinder within a cube.**

**Diagrams 5 - 6 show that when a three-dimensional cylinder is rotated so that its lateral length is facing us, we can envision the sphere within the cylinder.**

**From this perspective, it becomes apparent that by removing the four corners of the cylinder, 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 within a 16 cm x 16 cm wooden cube.**

**To form the cylinder from the wooden cube; we placed it on a wood lathe, and then rotated the cube and shaved away the four lateral corners, which are equal to one-quarter of the mass of the cube. This left a three-quarter mass of**

**the cylinder remaining.**

**The two diagrams above demonstrate the following: when given an angled frontal aspect and turned towards us, while at the same time losing its dimension of curvature, the cylinder, in effect, becomes a flat square.**

**It is then apparent that the removal of the four corners of the square will, in effect, remove one-quarter of the mass of the cylinder, releasing the potential sphere within.**

**Diagrams 1 - 5 depict the potential cylinder within a cube.**

**Diagrams 5 - 6 show that when a three-dimensional cylinder is rotated so that its lateral length is facing us, we can envision the sphere within the cylinder.**

**From this perspective, it becomes apparent that by removing the four corners of the cylinder, the shape of the sphere will be released.**

**Therefore**

**As a cylinder is three-quarters of its cube, so a sphere is three-quarters of its cylinder.**

**The diagram above depicts the three-quarter area of the circle of the square, relative to the potential cylinder within a 16 cm x 16 cm wooden cube.**

**To form the cylinder from the wooden cube; we placed it on a wood lathe, and then rotated the cube and shaved away the four lateral corners, which are equal to one-quarter of the mass of the cube. This left a three-quarter mass of the cylinder remaining.**

**The two diagrams above demonstrate the following: when given an angled frontal aspect and turned towards us, while at the same time losing its dimension of curvature, the cylinder, in effect, becomes a flat square.**

**It is then apparent that the removal of the four corners of the square will, in effect, remove one-quarter of the mass of the cylinder, releasing the potential sphere within.**

**The first diagram above serves to depict the cube fixed on a wood lathe before its four corners are carved away so as to form the cylinder. The second diagram depicts the cylinder placed lengthwise and laterally away from us, before the corners of the circular face, are shaved off so as 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.**

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

**In sum, regarding 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 cylinder is three-quarters of its cube.**

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

**CONFIRMATION BY MASS**

**Given that the cube weighed 160 grams before 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**

FROM A CUBE TO ITS SPHERE

FROM A CUBE TO ITS SPHERE

**CUBE TO ITS CYLINDER**

**For the sake of mathematical ease, we are going to use a cube which measures sixty centimetres.**

__Steps__**1. Measure the length of one right-angle to obtain a “diametric distance” of 60 cm's.**

**2. Multiply the diametric distance of 60 cm's by 60 to obtain the 360 square cm's, to the area of one face of the cube cube, and a square perimeter length measuring 24 cm's.**

**3. Multiply the 360 square cm's area of one face of the cube by 60, to obtain the cubic capacity of 21,600 cubic cm's to the volume of the cube.**

**4. Divide the cubic capacity 21600 cubic cm's by 4, to obtain 5,400 cubic cm's, which is one-quarter of the cubic capacity of the cube.**

**5. Multiply the 5,400 cubic cm's by 3, to obtain 16,200 cubic cm's. which is the cubic capacity of the cylinder.**

**6. Multiply the 360 square cm's area to one face of the six-sided cube by 6, to obtain the cubes overall surface area of 2,160 sq cm.**

**7. Divide the cubes overall surface area of 2,160 square cm's by 4, to obtain 540 square cm's, which is one-quarter of the cubes overall surface surface area.**

**8. Multiply the one-quarter surface area 540 square cm's by 3, to obtain 1,620 square cm's, which is the square area of the overall surface area to the cylinder.**

__Sum__

The cubic capacity of the cylinder is 16, 200 cubic cm's, which is three-quarters that of the 21,600 cubic cm's to the volume of the cube.

The surface area of the cylinder is 1,620 square cm's, which is three-quarters that of the 2,160 square centimetres to the overall surface area of the cube.

The number of square cm's to the overall surface area of the cylinder is 1,620 square cm's, which is one tenth of the number of cubic cm's to the volume of the cylinder, which is 16,200 cubic cm's.

The cubic capacity of the cylinder is 16, 200 cubic cm's, which is three-quarters that of the 21,600 cubic cm's to the volume of the cube.

The surface area of the cylinder is 1,620 square cm's, which is three-quarters that of the 2,160 square centimetres to the overall surface area of the cube.

The number of square cm's to the overall surface area of the cylinder is 1,620 square cm's, which is one tenth of the number of cubic cm's to the volume of the cylinder, which is 16,200 cubic cm's.

**CYLINDER TO ITS SPHERE**

**9. Divide the Cylinders cubic capacity of 16,200 cubic cm's by 4, to obtain 4,050 cubic cm's. which is one-quarter of the cubic capacity of the Cylinder.**

**10. Multiply the 4,050 cubic cm's by 3, to obtain 12,150 cubic cm's, which is the volume of the sphere, and three-quarters of the volume of its cylinder.**

**11. Divide the Cylinders overall surface area of 1,620 square cm's by 4, to obtain 405 square cm's which is one-quarter of the 1,620 square cm's, to the surface area to the Cylinder.**

**12. Multiply the one-quarter surface area of 405 square cm's by 3, to obtain 1,215 square cm's, which is the surface area of the sphere, and three-quarters of the 1,620 square cm's to the surface area of its cylinder.**

**FROM THE CYLINDER TO ITS CONE**

**CONE**

**THREE-DIMENSIONAL - SOLID PERSPECTIVE TWO - DIMENSIONAL - FLAT PERSPECTIVE**

**In our previous work Twelve Steps To The Sphere we determined that a cylinder has a surface area and a cubic volume that is is three-quarters that of its surrounding cuboid.**

**In the two diagrams A and B above of a three-dimensional cone and the cone within its cylinder, it can be seen that the volume and the triangular shape of the cone represent half of the volume and half of the shape of the surrounding cylinder.**

**In the two diagrams C and D above of a frontal straight forward two-dimensional view of a cone and the two- dimensional view of the cone as it would be within its cylinder.**

**It can be seen that if the two central upright half-triangles of the cylinder were given the three-dimensional aspect of a cone.**

**And the two outer downward pointed triangles of the cylinder were joined together to form an upright half-triangle of the cylinder, and this then given the three-dimensional aspect of a cone.**

Each of the two cones would have half of the surface area and half of the volume of the cylinder.

Each of the two cones would have half of the surface area and half of the volume of the cylinder.

**Therefore it follows that as a cone has a volume and surface area that is half of its surrounding cylinder, and the cylinder has a volume and surface are that is three- quarters that of its surrounding cuboid, the volume and surface area of a cone is three-eighths that of the cuboid surrounding the cylinder.**

**THE AREAS AND VOLUMES**

**OF**

**OVALS AND OVOIDS**

**An Ovoid Is A Squashed Sphere - An Oval Is A squashed Circle**

**A flat-downward pressure applied to the center of a round ball at rest on a flat surface will cause the air within the ball to be displaced equally away from the vertical axis of the ball, due to the passive resistance of the surface beneath the ball.**

**The vertical compression of volume within the ball, in effect, causes the envelope of the ball to expand laterally and change its shape to that of an ovoid.**

**As space and area have interchangeable values of dimension, when the area within the envelope of a circle is vertically compressed, it will follow the same rule and transform into an oval.**

**Therefore if the two lateral and vertical diametric distances (diameters) are added together to give a single length, and this single length is then divided into two equal lengths.**

**Each of the two**

**equal**

**lengths can be equated as to being two equal diametric distances (diameters) of either a circle or a sphere.**

And as such, the same methods we have used to find the area to a circle, and to find the surface area and volume of a sphere, can be applied to whichever the case may be.

And as such, the same methods we have used to find the area to a circle, and to find the surface area and volume of a sphere, can be applied to whichever the case may be.

**Therefore we can say**

**As a circle is three-quarters of its square, so its oval is three-quarters that of its oblong-rectangle**

**As a sphere is three-quarters of its cylinder, so its ovoid is three-quarters that of the extended cylinder.**

**THE NUMBER**

OF

VERTICALLY AND HORIZONTALLY ORIENTATED DEGREES

OF

TWO-DIMENSIONAL LATITUDE AND LONGITUDE

OF

VERTICALLY AND HORIZONTALLY ORIENTATED DEGREES

OF

TWO-DIMENSIONAL LATITUDE AND LONGITUDE

**TO**

THE CURVED SURFACE AREA OF A SPHERE

THE CURVED SURFACE AREA OF A SPHERE

**(Based upon there being 360-degrees to the Edge of a Circle)**

****

__Reminder__

Using a 120-centimeter length diameterUsing a 120-centimeter length diameter

**1. Multiply the 120-centimeters by 3**

**2. The Circles length is 360-**

**centimeters**

**3. Every Circle has 360-degrees**

**The Circles length is 360-centimeters, 1 degree is 1-centimeter long**

**4. Multiply the 120 centimeters by 4, the length of the circles square is 480-centimeters**

** The Circles length is three-quarters the length of the circles square**

****

**Square degree - Wikipediahttps://en.wikipedia.org/wiki/Square_degree**

Analogous to one degree being equal to π180 radians, a square degree is equal to (π180)2, or about 13283 =3.0462×10−4 steradians (0.30462 msr). The number of square degrees in a whole sphere is approximately41253 deg2.

More simply this decimal and π ***approximation*** is stating that a radian is an arc of a circle that is equal in length to that of its radius, rather than as we now know, the length of a radius and thus also a radian, is equal to exactly one-sixth of the length to that of a circles edge.

Analogous to one degree being equal to π180 radians, a square degree is equal to (π180)2, or about 13283 =3.0462×10−4 steradians (0.30462 msr). The number of square degrees in a whole sphere is approximately41253 deg2.

More simply this decimal and π ***approximation*** is stating that a radian is an arc of a circle that is equal in length to that of its radius, rather than as we now know, the length of a radius and thus also a radian, is equal to exactly one-sixth of the length to that of a circles edge.

**Method**

Essentially a circle represents a single cross section of a sphere, therefore a 360-degree marked circular protractor can be used to represent a single cross section of a sphere.

Essentially a circle represents a single cross section of a sphere, therefore a 360-degree marked circular protractor can be used to represent a single cross section of a sphere.

**When viewed from a frontal vertical aspect, the edge to the circle of the protractor has 360-degrees to its length, and there are 180 levels of degrees to its height.**

**Therefore it follows that when the circular protractor is rotated laterally to complete one full rotation of 360-degrees around its axis, each one of the 180 degrees to the circle of its height will have traveled 360-degrees, and as such in unison completed one full sphere of rotation.**

**Sum**

**180-degrees of circular height times 360-degrees of lateral rotation equals 64, 800 *two-dimensionally orientated***

*(Longitudinal and Latitudinal)*degrees of *curvature* to the surface area of a sphere.

**THE AREA OF 1 DEGREE**

**OF**

**64, 800 DEGREES**

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

**⁰**

**of height to the circle of a protractor 360**

**⁰, which served t**

**o demonstrate that there are**

**64, 800**

**⁰**

**of longitude and latitude to the curved surface area of a sphere.**

**However, this does not serve to give each of those degrees an equal amount of surface area.**

**The reason being, each subsequent circle of 360**

**⁰**

**to that of the circle's equator, is smaller than the previous circle.**

**To equate the areas of all of the degrees to the surface of the sphere, we divide the number of 64,800 degrees into the overall surface area of the sphere.**

**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 square cm ÷ by 4 = 21, 600 square centimeters to one-quarter of the surface area of the cube.**

**21, 600 square cm x 3 = 64, 800 square centimeters to the surface area of the ¾ cylinder of the cube.**

**64, 800 square cm ÷ by 4 = 16, 200 square cm to one-quarter of the cylinder's surface area.**

**16, 200 square cm x 3 = 48, 600 square centimeters to the surface area of the ¾ sphere of the cylinder.**

**48, 600 square 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 plus three times 1/3rd rotation of 120 degrees (360) = 64, 800 degrees.**

**Proving once again; that a circle's length is 3 x its diameter length and its area is 3/4 that of its surrounding square.**

**THE**

NUMBER OF VERTICALLY HORIZONTALLY AND DIAGONALLY ORIENTATED DEGREES

TO

THE CURVED SURFACE AREA OF A SPHERE

NUMBER OF VERTICALLY HORIZONTALLY AND DIAGONALLY ORIENTATED DEGREES

TO

THE CURVED SURFACE AREA OF A SPHERE

**(Based upon there being 360-degrees to the Circle)**

**Bearing in mind as previously, that a circle essentially re**

**presents a single cross section of a sphere, and therefore a 360-degree marked circular protractor can considered as to being representative of a single cross section of a sphere.**

As we know and as can e observed in the diagram of the protractor above, a circle has 360 radii which extend *radiate* outwards from its centre and extend into its surrounding square, and as such both the circle and the square have the same number of degrees to their makeup.

And so it follows, that if we were to rotate

As we know and as can e observed in the diagram of the protractor above, a circle has 360 radii which extend *radiate* outwards from its centre and extend into its surrounding square, and as such both the circle and the square have the same number of degrees to their makeup.

And so it follows, that if we were to rotate

**the circular protractor and its surrounding square laterally to complete one full rotation of 360-degrees around its axis, the circle in ***transitional effect*** would become a sphere, its square would become a cube, and its radii would become**

**radon's**

**.**

**Therefore as the cube of the sphere has six square faces/aspects to its make up, so then also does the sphere within its makeup.**

**Given this, if we add the 360-degrees to each of the six faces/aspects together, the number of degrees to both the sphere and its**

**surrounding**

**cube is 6 times 360-degrees = 2 160 degrees.**

**The immediate question which springs to the discerning mind, is why does the sphere have 68 800 longitudinal and latitudinal or so-called square degrees to its**

**surface, and yet only 2 160 degrees to the six faces/aspects of its surface?**

**And the reason is, that there is no such thing as a square degree or of any other geometric shape to a degree, when considering a sphere in regard to its longitudinal and latitudinal (vertical and horizontal) aspect, we are projecting a flat two- dimensional square grid system over its curved surface area.**

**However when we consider the sphere from a three-dimensional aspect, we are relating to the number of radii/radon's/degrees extending/radiating from the spheres centre toward its outer surface.**

**And please note, we will be covering the transitional nature of degrees relating to three-dimensional thinking, rather than Euclidean thinking later, because as is always the case I have to take the time to find the right wording to simplify and try to translate my thoughts into a format that can be easily understood by the reader.**

I had hoped in regard to the previous paragraph quote;

__Disappointment__I had hoped in regard to the previous paragraph quote;

*****Therefore as the cube of the sphere has six square faces/aspects to its make up, so then also does the sphere within its makeup.**

**Given this, if we add the 360-degrees to each of the six faces/aspects together, the number of degrees to both the sphere and its**

**surrounding**

**cube is 6 times 360-degrees = 2, 160 degrees***.**

**That some readers would have**

**immediately jumped on this quite outrageous statement, and then Emailed to inform me that as a sphere has 64, 800 two-dimensionally orientated vertical and horizontal degrees to its surface area, it is totally ludicrous for me to claim that there are only 2 160 degrees to the surface area of a sphere, which has an additional third-dimension of orientation.**

**If we consider the six faces of a cube relative to the 360-degree circle of the protractor, they are six flat two-dimensional aspects of the cube which as such and as previously explained equate to six cross sections of a sphere, and if conjoined at their centers the number of radon's radiating outwards from the centre of the sphere would only amount to 2, 160 radiant degrees (radon's) rather than the correct number of 45, 656000 degrees.**

**Just as there are 360 equally spaced degrees to the length of the flat two-dimensional perspective of a circles edge, so it follows that all of the three-dimensionally orientated degrees to a spheres surface will also be equally spaced, and each degree possess the same amount of the spheres surface area as all of the other three-dimensional degrees of orientation.**

To find the equally spaced square area of a larger square, we multiply the length of one side of the square by itself

*, and we then multiply this result by the same amount of length to find the equally spaced square surface area of a three-dimensional cube.*

**(reminder, and as we know the area of the squares circle area is three-quarters that of its square, a cylinder is three-quarters of its cube and a sphere is three-quarters of its cylinder )**Therefore as we are dealing with the number of three-dimensionally orientated degrees to the surface area of a sphere, and as each degree is equal in area and each shares the same amount of overall area to the spheres surface, so it follows that the number of degrees to the surface of the sphere, will equate to the same number of degrees to that of the squared surface area of the cube.

Sum

360 x 360 x 360 = 45, 656000 squares to the surface area of a cube, and the same number of degrees to both the surface area of a cube and the surface of the potential sphere within the cube.

THE IRRATIONALITY

OfTHE IRRATIONALITY

Of

**π**

**Oxford English Dictionary (OED)**

**RATIO: the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.**

**Pi: symbol of the "ratio" of the circumference of a circle to its diameter, approximately 3.14159**

**IRRATIONAL: Mathematics (of a number, quantity, or expression) "not expressible as a ratio" of two integers, and having an infinite and non-recurring expansion when expressed as a decimal.**

**Examples are the numbers pi and the square root of 2.**

........

**Quite clearly, these two definitions regarding pi contradict each other. The reason is as follows.**

**On the one hand, pi is defined as being;(as quoted) the symbol of "the ratio" despite the fact that pi does not have a ratio, as it cannot be equally divided into either a circle's length or its circumference.**

**On the other hand, pi is defined as being;(as quoted) an "irrational number." However, it is not. When observed in its original form as the improper fraction of 22/7, which equates to 3 1/7, it is blatantly clear that pi is not a number but rather three whole (3 x 1) units of one diameter length, with 1/7th of one diameter length left over.**

**Or, more simply, π is three and a bit.**

**Therefore, the greatest irrational number that exists in the fields of geometry and mathematics, is the number of geometers and mathematicians who irrationally keep trying to use three and a bit (3 & ?) to approximate a circle instead of using the Sumerian rational number 3, which is three whole units which subdivide a circle's edge into three identical lengths.**

**And insist that the diameter of a circle is its circumference line divided by pi.**

**Oxford English Dictionary**

**Circle: noun. a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center) ... 'In the normal geometry of flat space, the diameter of a circle is its circumference divided by pi.'**

........

**This definition of a circle is incorrect.**

**Author's Definition of a Circle**

**Circle: noun. 1. A round shape whose size is defined by the length to its circular continuum of edge, which consists of 360 radial points of reference called degrees, all of which are equidistant from a fixed central point of circumnavigation.**

**A round shape whose circular continuum of an edge is defined by the visible contrasts of color and texture, as opposed to the color and texture of its limiting surrounding medium.**

**The diameter of a circle is the length of its circular edge continuum, divided by 3.**

**For more than 2000 years since the times of Euclid and Archimedes, it has been universally taught and accepted that shapes and solid bodies possess outlines, but they do not! outlines do not have any more existence than the imaginary linear patterns of the star constellations.**

**Primary shape example;: the Circle.**

**A circumscribed (drawn) circle has a round area of a surface to its round form, with a circular area of graphite or ink surrounding its area of the circular form.**

**A dinner plate is solid round form with a circular edge, which is in direct abutment with the surrounding gaseous atoms of the atmosphere.**

**A cylinder is solid round form, which has a single straight dimension of height, and a round or oval shape in cross section; which is in direct abutment with the surrounding gaseous atoms of the atmosphere.**

**CIRCLE VERSUS CIRCUMFERENCE**

**TWO CIRCLES: ONE BLACK ONE ORANGE THREE CIRCLES: ONE BLACK ONE RED ONE ORANGE**

**Oxford English Dictionary**

**Circle: noun. a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center) ... 'In the normal geometry of flat space, the diameter of a circle is its circumference divided by pi.'**

**This definition of a circle is incorrect.**

**Author's Definition of a Circle**

**Circle: noun. 1. A round shape whose size is defined by the length to its circular continuum of edge, which consists of 360 radial points of reference called degrees, all of which are equidistant from a fixed central point of circumnavigation.**

**A round shape whose circular continuum of an edge is defined by the visible contrasts of color and texture, as opposed to the color and texture of its limiting surrounding medium.**

**The diameter of a circle is the length of its circular edge continuum, divided by 3.**

**For more than 2000 years since the times of Euclid and Archimedes, it has been universally taught and accepted that shapes and solid bodies possess outlines, they do not! outlines do not have any more existence than the imaginary linear patterns of the star constellations.**

**Primary shape example: the Circle.**

**A circumscribed (drawn) circle has a round area of a surface to its round form, with a circular area of graphite or ink surrounding its area of the circular form.**

**A dinner plate is a solid round form with a circular edge, which is in direct abutment with the surrounding gaseous atoms of the atmosphere.**

**A cylinder is a solid round form, which has a single straight dimension of height, and a round or oval shape in cross section; which is in direct abutment with the surrounding gaseous atoms of the atmosphere.**

****

****

*SYMMETRICAL RATIONALITY*

**Card A Card B**

**Card B Card A**

**We have two square yellow cards measuring 120 cm’s x 120 cm’s**

**Card A. has one black circle of 120 cm diameter and has been cut into four equal quadrants of 60 cm squares**

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

**A. Black Areas**

**All black areas have an equal area to each other****Any number or type of black area combined will give an equally divisible area****All black areas combined will give an equally divisible area**

**B. Yellow Areas**

**All yellow areas have an equal area to each other****Any number or type of yellow area combined will give an equally divisible area****All yellow areas combined will give an equally divisible area.**

**C. All Areas of The Two Cards**

**All areas of both cards are equal area.****Any number and any combination of black and yellow areas will give an equally divisible area.****All black and yellow areas combined will give an equally divisible area.****All areas of the two cards combined will give an equally divisible area (288 squares).**

**Logic**

**All of the black and yellow areas of the circle and square contain an equally divisible -and rational amount of area.**

**When an equally divisible -and rational amount of area is deducted from an equally divisible -and rational amount of area, an equally divisible -and rational amount of area remains.**

**The formula for calculating pi, however, always equates to an irrational -and unequal amount of area (of the line) that cannot be equally subdivided into the length or the area of a circle.**

**ALPHA AND OMEGA**

INFINITY AND ETERNITY

INFINITY AND ETERNITY

**Given a singular round shape, the circular length of that round shape, exists as one of any number of an**

***infinity* of possible lesser or greater sized concentric circles.**

**Given that a singular round shape exists of itself as one circular length of an**

***infinity* of possible lesser or greater sized concentric circles, the length of a circle**

**does not possess any *physical or numeric value* unless it has been ascribed either or both of these by an intelligence.**

**.**

**In this regard as we are aware, all circles were designated by the ancient Sumerians as to**

**possessing, *0 to 360 equally spaced points of degree, equidistant from the center of the circle's round shape.**

**INIFINITY**

**All Circles have 0 to 360 equally spaced equidistant degrees to their length.**

**The 0 Degree spatial position on a circles length, is one and the same 360 Degree spatial position on the circles length.**

**Essentially 0 Degrees is 360 Degrees and 360 Degrees is 0 Degrees.**

**Therefore**

The 0 or Alpha Degree position which begins the Circle, is also the 360 or Omega Degree positon which ends the Circle, and vis-a-vis.

The 0 or Alpha Degree position which begins the Circle, is also the 360 or Omega Degree positon which ends the Circle, and vis-a-vis.

**And as the Alpha Degree is the Omega Degree and the Omega Degree is the Alpha Degree, as such it entirely endless.**

**The circle has no beginning or end, as its beginning is its end and its end is its beginning, ad infinitude.**

**And as the Alpha Omega point of a circle or a cycle, can exist anywhere along the physical length of a circle and any-when along the length of an energetic cycle,**

So it follows that the physical energy of an infinite circle or a cycle, can only exist in conjunction with the eternal space that allows its infinite existence.

And as this universal infinite and eternal cyclic state of all energy, mass and space, has been empirically and unequivocally proven to be beyond any doubt by the sciences.

It has been laid down as a fundamental law of physics which states,

Energy may be converted, but energy cannot be created nor destroyed.

More simply: As energy is infinitely convertible from one form of energy to another and vis-a-vis it is indestructible, and as the universe is comprised of energy, mass and space; so it follows that the universe of energy, mass and space is indestructibly infinite and eternal.

So it follows that the physical energy of an infinite circle or a cycle, can only exist in conjunction with the eternal space that allows its infinite existence.

And as this universal infinite and eternal cyclic state of all energy, mass and space, has been empirically and unequivocally proven to be beyond any doubt by the sciences.

It has been laid down as a fundamental law of physics which states,

Energy may be converted, but energy cannot be created nor destroyed.

More simply: As energy is infinitely convertible from one form of energy to another and vis-a-vis it is indestructible, and as the universe is comprised of energy, mass and space; so it follows that the universe of energy, mass and space is indestructibly infinite and eternal.

THE

TWELVE SQUARE

THE

TWELVE SQUARE

**PHENOMENON OF THE PYTHAGOREAN THEOREM**

**PYTHAGORAS 2582 - 2500 BC**

**In**

**any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.****www.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 Pythagorean theorem appears on the Babylonian Susa Tablet dated between, 1,800 - 1,600 BC.**

**A B**

In Diagram A, it can be seen that given a right-angle triangle with a base and vertical line measuring 12 squares, the square on the hypotenuse measures 17 x 17 squares, which equates to 289 squares.

In Diagram A, it can be seen that given a right-angle triangle with a base and vertical line measuring 12 squares, the square on the hypotenuse measures 17 x 17 squares, which equates to 289 squares.

**However, when the number of squares and half -squares of this "45-degree" angle square on the hypotenuse, is counted; there are 264 whole squares + 48 half squares = 288 whole squares, to the "45-degree" square on the hypotenuse.**

**In Diagram B of the 17 x 17 "vertical" square, it can be seen that there are 17 rows of 17 squares,which amount to 289 squares, which is one square more than the sum of the squares on the other two sides of the right-angle triangle.**

**Therefore, given all of theproof that has been given regarding the Pythagorean theorem over previous millennia, one can only assume that this phenomenon has been missed or ignored and placed in the “too hard” basket.**

**DECIMAL MONETARY SYSTEM**

THE

Fraudulent

Decimal And Digital Decimating Arithmetic

of

Zionist - Grecian - Roman

Weights And Measures And Banking

AKA

Capitalism

THE

Fraudulent

Decimal And Digital Decimating Arithmetic

of

Zionist - Grecian - Roman

Weights And Measures And Banking

AKA

Capitalism

**Quote: USA Gangster Al Capone.**

Capitalism Is The Racket Of The Ruling Classes.

Capitalism Is The Racket Of The Ruling Classes.

**Quote: John Maynard Keynes**

Capitalism is the extraordinary belief, that the nastiest of men for the nastiest of motives, will somehow work together for the benefit of all.

Capitalism is the extraordinary belief, that the nastiest of men for the nastiest of motives, will somehow work together for the benefit of all.

__Wages and Salaries__**On an international scale, without any consultation or choice; we the working people's of the world blue and white collar alike, have "been forced " by the Big Banks and Business Corporations and their puppet governments, to receive all of our wages and salaries, "not directly" from our employers as once was the case, but rather "second hand" via the Banks.**

**Every dollar earned by billions of honest working people's worldwide has to be handed over to the loan sharking world of the Banking industries; which collectively, amounts to their receiving an unearned and tax-free income of trillions of salary/wage/ payday dollars, of every worker in the Banking and Marketing Big Brother run capitalist world in perpetuity.**

**And secondary to this, by making loans back to us from the massive pools of the trillions of our wage packet dollars they have stolen from us, they then make trillions of more dollars limited only to the amounts of interest they choose to charge us, for us borrowing back some of the money from the massive pools of money they have stolen from us.**

**However the cruellest irony of all is that they charge us Bank charges for their accounting of the humongous interest profits they have made, on the trillions of dollars they have stolen from us and then charged us interest on.**

**And perhaps as a final irony and insult they charge us ATM fees every time we make a withdrawal to regain back part of the money they steal from us every payday.**

**Worst example of Banking theft and interest fraud: In February 1986 having as with tens of thousands of others who had been previously lied to and conned into coming Australia, I had to set up an Australian Bank account to receive my salary.**

**On setting up a savings account with the Commonwealth Bank the following day, I asked what the rate of interest was for the savings account; she looked shocked and exclaimed Interest!?, I said yes what the rate of interest on a savings account. She then replied oh no there is no interest paid on savings accounts, and there never has been.**

**I was astounded, in the UK at that time the interest was for a Post Office savings account 5% Banks around 6 to 7.5% Building Societies around 10 to 11%; and here in Australia unbelievably absolute zero nothing.**

**And up until this day 33 years later it remains the same, except for accounts consistently containing a few thousand dollars whereby a minimal amount of interest is paid to the victims of their dishonest dealings.**

**Fourthly these vast pools of our stolen money, are then loaned to and made use of by their Big Business Incorporated Counterparts (Mates), to purchase our tax-payer funded public owned utilities, our common held lands, and our natural resources on the cheap**

*(so-called privatisation)*

**from our corrupt governments.**

With the politicians who have been involved in aiding and abetting these Big Corporations in privatising-stealing "our common-wealth", being assured by the those Corporations:

With the politicians who have been involved in aiding and abetting these Big Corporations in privatising-stealing "our common-wealth", being assured by the those Corporations:

**That when their lucrative political careers are over, on top of receiving millions of dollars in superannuation, and tax payer funded pensions with gold travel cards for life, they will also be given a highly paid consultancy, in one of the Corporations companies.**

**Adage**

**Look After The Pennies And The pounds Will Take Care Of Themselves**

And since the minting of the first coins by the Greeks in Lydia in 600 BC

And since the minting of the first coins by the Greeks in Lydia in 600 BC

*(Beware Greeks bear*

*i*

**ng gifts)****.**

**this has certainly proven to be the case.**

__Decimal & Digital Monetary - Weights and Measures__

Modern DayModern Day

**+ 1 Cent**

**Marketers and retailers who advertise an item as costing 99 cents, are liars, who use this ploy as a means of conning us into believing that we have paid less for a purchase, than the 100 cents to the dollar we are forced to pay them.**

**Whereas, under the old imperial system of 240 pennyweights of gold to the pound, advertising an item as costing 239 pence instead of 240 pence, would have more obviously been seen to be a ploy, and the smaller amount hardly worthwhile for customer and retailer alike.**

**+ 2 Cents**

**Given the computerized rounding up and down which takes place with purchases costing between 95 and 100 cents; the integrated computer programs of super market retailing, always ensure that over each section and the full range of their goods and products, it is always the customer who is robbed of the odd two cents.**

__In Reality__**99 cents does not constitute the total, or the the full value of 100 cents to a dollar unit.**

**&**

**0.9999999999... does not constitute the total, or the full value of one whole numerical unit of work/energy. or one whole unit of anything.**

__Questions & Answers__Decimal - Digital

__Versus__Empirical - Imperial__Decimal__**Using a calculator or a computer, divide 100 cents to the dollar by 3, this will yield an answer of 33.3 cents recurring.**

Multiply 33.3 cents by 3 this will yield an answer of 99.9 cents recurring.

Multiply 33.3 cents by 3 this will yield an answer of 99.9 cents recurring.

**Question:**

**Where did the 0.1 of a cent that you lost go?**

Answer: Into the Banks - Money Lenders own accounts.

Answer: Into the Banks - Money Lenders own accounts.

**Question:**

**When an ounce of gold or goods are weighed on a digital (decimal) scale, where does the missing 0.1 of an ounce go?**

**Answer:**

**Into the goldsmith's or the retailers pocket.**

**THE CASE AGAINST DECIMALISATION**

A.C.Aitken

Formerly Professor of Mathematics at the University of Edinburgh.

Introduction

It has long been known to mathematicians that the system of numeration which, by gradual evolution, we have inherited from previous ages and now use, namely the decimal system, is not the ideal system.

Equally it has been known that there has always existed a superior system, the duodecimal, certainly possessing some defects — since no system can be perfect — but superior in all important respects to the decimal system.

The great names in the list of those who have explicitly criticised the decimal and upheld the duodecimal are:

Blaise Pascal, that outstanding mathematical and religious genius of the early seventeenth century.

Gottfried Wilhelm Leibniz, philosopher and theologian, joint inventor with Newton of the differential calculus, first of all names in perceiving the possibility of expressing logic itself in mathematical terms and notation.

Pierre Simon Laplace, the celebrated mathematician of the later eighteenth and early nineteenth century, expositor of celestial mechanics, founder of the modern mathematical theory of probability, a name still associated with formulae and methods which are household words in mathematical analysis.

Pascal in 1642 at the age of nineteen invented an adding machine.

Leibniz in 1673 at the age of twenty-seven exhibited an adding-and-multiplying machine at the Royal Society of London.

As for Laplace, he is related to our topic by the fact that with Borda, Condorcet, Lagrange and Monge he was one of the Commission set up by the French Academy of Sciences in 1790 to examine the possibility of a decimal system of metric and of currency, and to take steps to introduce it. It is known that in the early stages of these deliberations the possibility of a duodecimal system, recognised as superior to the decimal, was discussed; but that it was rejected, on the ground that it was out of the question to educate the French public, within reasonable time, in this kind of calculation. In Britain, where the dozen had more uses, these considerations might have weighed less.

At any rate decimal currency was imposed on France in 1795, and the metric system, which ought logically to have preceded or been simultaneous with the currency change, since commodity and its measurement logically precede the monetary medium, was postponed until 1799. This however was not intentional; both changes would have been made together but that the quadrant of the Earth had to be accurately surveyed (as was done by measuring an arc between Dunkirk and Barcelona), and this difficult piece of geodesy could not be completed before 1799. Only then could the standard metre be adopted.

When all was over, regrets were felt by some, not then but later. Laplace himself in his later years gave expression to these; and one can hardly doubt that when, in his last recorded words as uttered to his disciple Poisson, “L’homme ne poursuit que des chimères”, he included, among those phantoms captured and found wanting, the decimal and metric system.

Napoleon himself (Napoleon’s remark was characteristic:“Twelve as a dividend has always been preferred to ten. I can understand the twelfth part of an inch, but not the thousandth part of a metre”) expressed regret for the extirpation of the number twelve from numeration and from exchange, for that is what any proposal of wholesale decimalism implies.

It implies indeed, as will be shown in cumulative detail later in this essay, the elevation to an undeserved place of a very unsuitable integer, namely ten, whose only distinctive property is that it divides by five, with the consequent demotion of twelve, a number divisible by 2, 3, 4 and 6, while its square, the gross, 144, divides by these and in addition by 8, 9, 12, 16, 18, 24, 36, 48 and 72, with all the consequences of economical and suitable use in parcelling, packaging, geometrical and physical construction, trigonometry and the rest, to which any applied mathematician and for that matter any practical man, carpenter, grocer, joiner, packer could bear witness.

Once again, currency should come afterwards and subserve all these; it should be in a one-to-one correspondence with them, which is indeed the reason for the traditional and well grounded British preference for the shilling with twelve pence, the foot with twelve inches; and also for the relation of the foot to the yard, since the number three, so intractable in the decimal system (consider one-third, 0.33333 . . ., or the similar equivalents for a sixth, a twelfth and the rest), precedes the number five in order, use and logic.

The twenty shillings to the pound was a characteristically British (indeed not British but English) attempt at reconciliation and compromise, for the French used not so much ten as the score (e.g. quatrevingts, quatre-vingt-dix), and this accommodation of twenty as well as twelve produced our hybrid system of pounds, shillings and pence, the disadvantage of which is precisely that it is hybrid, and therefore does not lend itself, as the decimal system does, to a “place” and “point” system of numeration. (A suggestion for rectifying this defect will be given later in this essay.)

With all this, however, pounds and pence have an advantage which the franc and centime, dollar and cent, metre and centimetre, cannot possibly claim, namely the exceptional divisibility of the number 240. This in fact is one of those integers which mathematicians, in that special field called the “theory of numbers”, are accustomed to call “abundant”.

An abundant number is one that has more factors than any number less than it; other examples of small size are 12, 24, 36, 60,120, 360. The gross, 144, or twelve dozen, just misses abundancy, being excelled by 120. Compared with 120 and 144, even with 60, the number 100 is relatively poverty-stricken in this respect — which indeed is why the metric system is a notably inferior one; it cannot even express exactly for example the division of the unit, of currency, metrical or whatever, by so simple, ubiquitous and constantly useful a number as three.

We are therefore entitled to ask: why, in this age of scientific progress, do we endure a system of numeration with so many disadvantages?

The answer removes us at once to remote history and probably prehistory; men counted on their fingers, and to this alone, reinforced, it is to be feared, by the indolent, unreflecting, and often arithmetically illiterate force of habit, the survival of the decimal system is due.

This cannot however last; men will not always evade decision by the facile and procrastinatory cliché of our times, “not practicable in the foreseeable future”.

In later paragraphs it will be indicated how new kinds of electronic computers, and the new type of education that this will enforce in the schools, universities and colleges of technology, are bound to produce a full acquaintance with four systems of numeration at least:

(i) the binary, based on two, the foundation of all electronic computation, to the exclusion (meanwhile) of the decimal except at the final stage of conversion and recording results;

(ii) the octonary, the system based on eight, by which binary results may by the simplest of transformations be compressed and held in store;

(iii) the decimal, since unfortunately, with all its defects, it is still with us;

(iv) the duodecimal, which in the opinion of many such as the writer will prove to be that system which translates the binary to the world at large, the world of men and women behind counters, ticket offices, carpenters’ benches, in stores, in homes.

History of Numeration

With such various introductory remarks, let us look at the history of numeration. We know of course, arithmetic in primitive times being necessarily primitive, that counting and barter were done on the fingers (whence the name digit for a number-sign), and that these hardened into written marks or into such movable objects as the beads or counters on the Chinese, Japanese or Russian abacus.

On the abacus, for example, the several parallel rods carrying counters are all crossed at right angles halfway along by a fixed dividing bar; each rod has on one side of the bar five counters, on the other side a single counter. (The number five, it is interesting to note, can be represented in two different ways; either, with the thumb, push all five counters up against the bar, or leave them alone and with the finger pull that other counter back against the bar.) The abacus, used by an expert, has remarkable resource and speed; during the American occupation of Japan, a Japanese with an abacus beat an American using a hand-operated calculating machine.

The whole point of mentioning this here is that if, for example, Russia should ever go duodecimal, a not unlikely possibility which would give her people, in all the ordinary calculations of life, an advantage of at least 35 manhours—so I reckon—in every 100, China could align herself with Russia even more simply, by having six counters instead of five on the half-rod of every abacus.

Ancient History

But to return to ancient history. The Sumerians of two thousand B.C., as is shown by certain cuneiform inscriptions brought to light not so long ago, used the ten system but also the sixty, the sexagesimal system; we have for example their multiplication tables. By 1800- 1700 B.C. something quite extraordinary takes place; the Babylonians take over from the Sumerians, and while still in the market place the scale of ten persists, the astronomers, architects, in fact what one may call the mathematicians, scientists, technologists of that remote period, the Hammurabi dynasty of 3700 years ago, constitute a hierarchy skilled in arithmetic to a degree unrivalled in the modern world; for they actually used the scale of 60, the sexagesimal scale, for fractions, reciprocals, even square roots. They have left the trace of their system in the 60-fold division of the hour into minutes and the minutes into seconds, a predominantly duodecimal subdivision, as one may see by looking at a clock, but in this we observe an accommodation not so much with the scale of ten as with 5.

Another such trace is the division of the whole circumference of the circle into 360 degrees. At the time of the French Revolution certain fanatical decimalists (following in the footsteps of Stevinus of Bruges two hundred years earlier) were for dividing the right angle into 100 degrees called “grades”, the half day into ten hours, even the year into ten months. These efforts, or rather the second of them, met with no success. Astronomers and surveyors will never use so defective a system; and numbers of instances can be cited, from trigonometry, periodic analysis, approximate evaluation of areas and volumes, and so on, in which a five-fold or ten-fold subdivision of the range gives formulae and methods remarkably inferior to a six-fold or twelve-fold one.

Those Babylonian mathematicians, by the way, have extensive tables, not only of reciprocals and square roots but actually of triads of integers making the sides of a right-angled triangle, the theorem of Pythagoras 1150 years before Pythagoras; but all in sexagesimal. The central point in all this is that 60 is an “abundant” number. That was why the Babylonians, masters of arithmetic in a way that, with certain exceptions, we are not, used it as a suitable base for their numerical system.

The Egyptians were not good at arithmetic; they could “ do sums”, but even the addition of vulgar fractions was carried out by them in an unbelievably cumbrous manner.

The Greek system of numeration was an inconvenient one, letters of the alphabet being used for numbers.

The Roman was hardly better, except that with a special kind of abacus they used a duodecimal notation for fractions, traces of which survive in two of our nouns, ounce and quincunx, that is to say, a twelfth and five-twelfths. For integers, however, they used the ten system and their well-known numerals; beautiful (none better, said Eric Gill) for lapidary inscriptions and coins, of no use for convenient calculation.

These endured in arithmetic almost up to A.D. 1500, simply because of the all-pervading dominance of the Roman Empire, and later of Rome itself.

In Asia this was not so; Hindu arithmetic had evolved special single symbols for the integers up to nine, together with the zero, long believed to be a Hindu invention until lately rediscovered, in an analogous role, in Babylonian cuneiform.

This Hindu system, with its excellent “place,’ convention, though not yet extended to fractional use with the “point”, percolated to Europe by way of the Arabs (for what we call Arabic numerals ought more justly to he called Hindu-Arabic), and the geography, early steps and manner of this percolation are worth a brief interlude.

Here it is convenient for speed to link in sequence a few sentences from Cajori’s History of Mathematics: “. . . at the beginning of the thirteenth century the talent and activity of one man was sufficient to assign to the mathematical science a home in Italy.... This man, Leonardo of Pisa, ... also called Fibonacci, . . was a layman who found time for scientific study.

His father, secretary at one of the numerous factories on the south and east coast of the Mediterranean erected by the enterprising merchants of Pisa, made Leonardo, when a boy, learn the use of the abacus. During extensive travels in Egypt, Syria, Greece and Sicily ... of all methods of calculation he found the Hindu to be unquestionably the best. Returning to Pisa he published, in 1202, his great work, the Liber Abaci, . . . the first great mathematician to advocate the adoption of the ‘Arabic notation’ “.

And later we read: “In 1299, nearly 100 years after the publication of Leonardo’s Liber Abaci, the Florentine merchants were forbidden the use of the Arabic numeral(s) in book-keeping, and ordered to employ the Roman numerals or to write the numeral adjectives out in full.” The interesting parallel, but in the opposite direction of legal enforcement of innovation, is that in 1801 and again in 1837 the French introduced legal penalties against those recalcitrants who still held out against the metric system.

Arabic Numerals

The system of Arabic numerals (really, as we have just seen, Hindu-Arabic) with its “place” convention — and this, not the choice of ten at all, is the real novelty and the real advantage — was thus introduced into Europe by one man, and had to fight its way for acceptance long years after he was dead. Thus a gravestone in Baden in 1371 and another in Ulm in 1388 are the first to show Arabic and not Roman numerals. Coins are more indicative: Swiss of 1424, Austrian 1484, French 1485, German 1489, Scots 1539, English 1551. The earliest calendar with Arabic figures is of date 1518. So our authority sets down; but he may be out in slight respects.

Napier and the Decimal Point

It would be tedious for the present purpose, however interesting for leisurely investigation, to pursue this. Enough to say that the first to invent the “decimal point”, written by him as a comma, was John Napier of Merchiston, in his Rabdologia of I6I7, the year of his death and three years after the publication of his logarithms.

Then in that era following the Renaissance, mathematics and arithmetic began to make the cumulative and ever-accelerating progress which we know; and so we come, by some drastic telescoping, to where this essay began, at the years 1790, 1795, 1799, the introduction of the metric system and the decimal system of currency, which Britain, having delayed so long with instinctive, characteristic and well-founded hesitation, is now considering.

I propose to vindicate in the ensuing paragraphs the soundness of that instinct, to show that Britain need adopt nothing whatever from France, America or the apparently progressive though in fact mathematically reactionary change of system in South Africa, and to try in some measure to forecast the future of computation.

The Duodecimal System

The episode of Leonardo Pisano is significant. The supersession of Roman numerals by Arabic digits, and eventually, but not all at once, by the “place” and “pointshifting” system, was in its initial stage the work of one man of perception but above all of conviction and energy.

This strength of conviction, but now in a new and even more progressive direction, namely that the system of Leonardo is not the final word but that the duodecimal system with appropriate notation is appreciably superior again, is held at the present time by a relatively small number of persons in the whole world. (It is true, of course, that the vast majority of the rest are entirely ignorant of the whole issue.)

One may mention the Duodecimal Society of America, counting in its membership distinguished actuaries and other prominent men— and it is symptomatic that such a society should take its origin in a country devoted since 1786 (a date in which America had no mathematical standing whatever) to decimal currency, though not, and this is again symptomatic, to decimal metric; there is a Duodecimal Society of Great Britain, recently founded, small in membership and resources; while in France, home of the decimal-metric system, there is M. Jean Essig, Inspecteur-Général des Finances, whose notable treatise on duodecimal arithmetic and measures, Douze: notre dix futur (Dunod I 955), is taken seriously, as the foreword shows, by Membres de l’Institut in France and Belgium. This small band of convinced men increases its numbers all the time and gains successes here and there, as when, for example, the most recent and progressive American school-texts on arithmetic and algebra, at the secondary stage, devote an extensive chapter to the description and appraisement of “scales of notation”, leaving the pupil in no doubt regarding the relative inferiority of the decimal system.

Yet anyone who enters into public discussion on duodecimal calculation comes at once upon the strangest circumstance. Incredible numbers of persons have been so imperfectly educated as to suppose that the decimal system is the only one that admits “place” notation and the property of shifting the “point” under multiplication or division by the base.

This defect of education, amounting in the case of certain newspaper correspondents to arithmetical illiteracy, has to be combated. The fact is that any integer whatever, suitable or unsuitable, can be taken as base of the corresponding system. A younger generation of persons selected by ability knows this already, namely all those who are preparing themselves for modern electronic computation, destined as it is, in the form of new machines not yet in production but easily imaginable, to transform in a hardly recognisable way whole domains of financial and official calculation, to say nothing of the arithmetical apparatus of technology generally.

For while1900-1925 was the period of the hand-operated mechanical calculating machine, and 1925 and onward that of the electrical one, from 1961 to the end of the millennium will be the era of electronic computers of every range, not merely of the large, and for certain purposes too large, ones that we see being installed in more and more places, but those of moderate size (and there will be smaller ones still) which are only now beginning to be in production. These will transform not merely arithmetic, but education in arithmetic; and a younger generation, familiar with binary and octonary systems as well as with decimal, will be sure to ask: What, reckoned in terms of time and efficiency, is the worth of the decimal system, and is there a better?

We shall without doubt see this happen, probably in Russia and America almost simultaneously, while we, who of all nations in the world are in the special and most favourable position to make the change, may be left behind; may well in fact have made a belated change, only to have to make a further belated one. Of course, on the other hand, there may be financial, economic and indeed political considerations which may enforce the other, to my mind reactionary, decision; but that would require a separate study, which has in some part been done and is in any case outside my competence. But I will simply say: political expediency is the ruin of science.

Monetary and Metrical Units

Why are we in that special and most favourable position? Because we already have the duodecimal system with us in all but name, and to a certain but lesser extent even in notation. I refer not to electronic machines, which can convert from their idiomatic binary into any other prescribed scale, but to the numberless transactions of ordinary life, in banks, ticket offices, behind counters, on board buses, wherever and whenever there is buying and selling and giving of change. Consider a railway clerk giving tickets and change, often at top speed to a heavy queue.

Does he ever think of decimal tables in handing back 5s. 7d. as change from a 10s. note on a ticket of 4s. 5d.? Not he; like hundreds of thousands of men behind counters he is a highly versed duodecimalist, though it would not occur to him to give so publicly useful a faculty so highsounding a name. I know this from having spoken recently with dozens of such men.

Here is a typical comment, from a Scots bus conductor: “We get on weel eneuch; yon would muck it all up again”. Some may think they might get on weel eneuch with decimal coinage; the most manage perfectly well. There is no cogent evidence that the public wish this change in the least; though the will of the public, strong as it might be either way, is neither the only nor the chief consideration. The French, at the very height, in 1790, of their enthusiasm for liberty, equality and fraternity, so qualified equality as to set up an academic commission of the most distinguished mathematicians in the land.

However, I propose — and it is not at all original with me — a certain change, a slight one, by which in a phased gradualness, an interregnum of years of quiet habituation and consolidation, we may bring in the more efficient system. It is: to have a pound, call it R for this discussion (a stag of twelve points is a royal!), of twelve shillings, a gross of pence.

It banishes at a stroke all oddments from twelve shillings and a halfpenny to nineteen shillings and elevenpence halfpenny; it is a paper note, a “royal”, that mediates between and supersedes the pound and ten-shilling note, requires no new minted coinage whatever, and is very close to one and two-thirds dollars. Call it then R1:0:0. Its half is R0:6:0 shillings. Its quarter R0:3:0. Its eighth R0:1:6. All very much as at present.

The half-crown might stay for a while, but eventually might be superseded by a three-shilling piece, a “quarter”, easier than the halfcrown to distinguish from the florin. Pennies and the rest are exactly the same as now. For example, except that we have this R of new value, we shall write R3:0:0; and the like as before.

So also for feet and inches. There might be — I do not know whether it is suitable or not, and would not presume to dictate to the practical measurer — a new “rod” simply of twelve feet, and this would make parallelism complete.

Duodecimalists should not dictate too much what is desirable; they may well leave it to practical craftsmen to find what is the best accommodation, provided only that the final outcome is indeed cast in a duodecimal hierarchy of units.

Here I differ from many duodecimalists; for I believe that, if the principle is once accepted, practical and intelligent men can be trusted to find possibly an even better solution than any duodecimalist or duodecimalist society might have proposed.

General Arithmetic However, to go further, let us pass from the monetary or metrical units and super- or subunits to the general arithmetic of the matter. Thus, let the fraction a half itself, in whatever context, be denoted by 0:6, a third by 0:4, a quarter 0:3, a sixth on:, twelfth 0:1, where the colon (most duodecimal publications use a semi-colon) serves for the duodecimal point, and will move right or left under multiplication or division by twelve.

For example, movement to the left. What is a twenty-fourth? A twelfth of a half, hence 0:06; a thirty-sixth is 0:04. A thirtysixth of the new royal is indeed fourpence; and so on. Contrast this with the inexact and inadequate third as 0. 33333 . . ., sixth as 0.6666 . . ., twelfth as 0.083333 . . ., and so on to more turgid examples.

Someone may say: What about a fifth or a tenth? Admittedly, since five does not go exactly into twelve, we shall here obtain a non-terminating duodecimal. For example, a tenth comes out as 0:12497 . . .. the last four digits forming the recurring period; but a close approximation to this is 0:125, committing the slight error, in excess, of 1/8640. (For comparison the approximation 0. 333 for one-third commits, in defect, an error of one three-thousandth.)

However, to go slightly further still. A shilling, 1:0s., is a dozen pence. Shift the colon to the right and in fact, since it is not then necessary, remove it, and write the dozen itself as *10, the prefixed asterisk (functioning like the American dollar sign) indicating that we are in a special system, that of the dozens, the meaning of the symbols being: one dozen, no units. Similarly thirteen, being one dozen, one unit, is *11; fourteen is *12, twenty-five is *21, and so on. The gross likewise is *100, meaning one gross, no dozens, no units; I will attend to names later. But all of this is just another way of writing 1:0:0 in the new R way, the kind of thing that faces us every day on a bill. Duodecimalism is nothing but this, though of course we have to know our tables, e.g. that 7 times 9 (asterisk with single-digit numbers not required) is *53, five dozens and three. But this is the smallest part, in a slightly different notation, of the first entries in any ready reckoner, and we have seen that already great sections of the population know these elementary tables, from habit, from serving customers and giving change. Consider the number, in decimal notation, 457. It is three gross, two dozen and one, *321. If these happened to be pence, then, in pounds, R3:2:1; in shillings, *32:1s., three dozen and two shillings and a penny. But this is to labour the habitual; we are doing this kind of thing all the time. Everyone who knows (some do not) that twelve articles at sevenpence each is seven shillings is simply saying that a dozen times seven is seven times a dozen, namely *10X7=*70 in pence, or in shillings *10X0:7s. = 7s. I showed some of this, doing some simple addition of fractions by it, to a bank teller and likewise to a stationer. The reaction was identical; each man involuntarily shielded his eye with his hand, doubtless to ward off the blinding flash of the obvious. Well, it is that some of this, in a differently couched and very uninspiring form, is taught in the chapter of school algebra dealing with “scales of notation”, though often treated in such a perfunctory fashion that the pupil may be excused from regarding it, as so much tediously useless manipulation. I exclude from my condemnation those admirable American school textbooks. We have suggested, provisionally, *10 for twelve, *11 for thirteen: for we hope eventually to use our system exclusively and to drop the asterisk. Confusion will be caused unless we devise new single symbols for ten and eleven; we can keep the names. Is it beyond the power of artistic typography (I suggest 7, an inverted 2, for ten; and 8, an inverted 3, for eleven) to invent simple, distinctive, cursive and aesthetically satisfying symbols for these two integers? The Hindus had to invent all ten of their symbols; while I could show many unsuspected situations in ordinary arithmetic where an alternative ten, at least, would have been valuable. On the Chinese and Japanese abacus there were and are two ways of expressing five, appropriate to different situations. For myself, I do my calculations with no great need for symbolic representation, but the above inversions of 2 and 3 served me well enough. Certain duodecimal societies, as well as a good many idiosyncratic individuals, have advocated various symbols, quite commonly X or c for ten, E for eleven, and so on. This will not do: letters of the alphabet must be kept for algebra, not arithmetic; let us think of the confusion in trying to write in such a way “ten times X”. So also for nomenclature.

For myself, I do not depend much on auditory impression for number, but thinking of the Scots”twal” I sometimes imagined “twel-one”, “twel-two”, and so on for thirteen, fourteen and the rest; but of course in dictation one would mention “asterisk” and call out, just as we do in decimal, “one one”, “one two” and the like for *12, *12, etc. There should be no difficulty here. Once again, duodecimalists should not prescribe too much for others in this matter; language and linguists should be able to find, as the French language does with never failing felicity, euphonious and idiomatic equivalents for any new entity that may arise. For example Icelandic also, when faced with the

A.C.Aitken

Formerly Professor of Mathematics at the University of Edinburgh.

Introduction

It has long been known to mathematicians that the system of numeration which, by gradual evolution, we have inherited from previous ages and now use, namely the decimal system, is not the ideal system.

Equally it has been known that there has always existed a superior system, the duodecimal, certainly possessing some defects — since no system can be perfect — but superior in all important respects to the decimal system.

The great names in the list of those who have explicitly criticised the decimal and upheld the duodecimal are:

Blaise Pascal, that outstanding mathematical and religious genius of the early seventeenth century.

Gottfried Wilhelm Leibniz, philosopher and theologian, joint inventor with Newton of the differential calculus, first of all names in perceiving the possibility of expressing logic itself in mathematical terms and notation.

Pierre Simon Laplace, the celebrated mathematician of the later eighteenth and early nineteenth century, expositor of celestial mechanics, founder of the modern mathematical theory of probability, a name still associated with formulae and methods which are household words in mathematical analysis.

Pascal in 1642 at the age of nineteen invented an adding machine.

Leibniz in 1673 at the age of twenty-seven exhibited an adding-and-multiplying machine at the Royal Society of London.

As for Laplace, he is related to our topic by the fact that with Borda, Condorcet, Lagrange and Monge he was one of the Commission set up by the French Academy of Sciences in 1790 to examine the possibility of a decimal system of metric and of currency, and to take steps to introduce it. It is known that in the early stages of these deliberations the possibility of a duodecimal system, recognised as superior to the decimal, was discussed; but that it was rejected, on the ground that it was out of the question to educate the French public, within reasonable time, in this kind of calculation. In Britain, where the dozen had more uses, these considerations might have weighed less.

At any rate decimal currency was imposed on France in 1795, and the metric system, which ought logically to have preceded or been simultaneous with the currency change, since commodity and its measurement logically precede the monetary medium, was postponed until 1799. This however was not intentional; both changes would have been made together but that the quadrant of the Earth had to be accurately surveyed (as was done by measuring an arc between Dunkirk and Barcelona), and this difficult piece of geodesy could not be completed before 1799. Only then could the standard metre be adopted.

When all was over, regrets were felt by some, not then but later. Laplace himself in his later years gave expression to these; and one can hardly doubt that when, in his last recorded words as uttered to his disciple Poisson, “L’homme ne poursuit que des chimères”, he included, among those phantoms captured and found wanting, the decimal and metric system.

Napoleon himself (Napoleon’s remark was characteristic:“Twelve as a dividend has always been preferred to ten. I can understand the twelfth part of an inch, but not the thousandth part of a metre”) expressed regret for the extirpation of the number twelve from numeration and from exchange, for that is what any proposal of wholesale decimalism implies.

It implies indeed, as will be shown in cumulative detail later in this essay, the elevation to an undeserved place of a very unsuitable integer, namely ten, whose only distinctive property is that it divides by five, with the consequent demotion of twelve, a number divisible by 2, 3, 4 and 6, while its square, the gross, 144, divides by these and in addition by 8, 9, 12, 16, 18, 24, 36, 48 and 72, with all the consequences of economical and suitable use in parcelling, packaging, geometrical and physical construction, trigonometry and the rest, to which any applied mathematician and for that matter any practical man, carpenter, grocer, joiner, packer could bear witness.

Once again, currency should come afterwards and subserve all these; it should be in a one-to-one correspondence with them, which is indeed the reason for the traditional and well grounded British preference for the shilling with twelve pence, the foot with twelve inches; and also for the relation of the foot to the yard, since the number three, so intractable in the decimal system (consider one-third, 0.33333 . . ., or the similar equivalents for a sixth, a twelfth and the rest), precedes the number five in order, use and logic.

The twenty shillings to the pound was a characteristically British (indeed not British but English) attempt at reconciliation and compromise, for the French used not so much ten as the score (e.g. quatrevingts, quatre-vingt-dix), and this accommodation of twenty as well as twelve produced our hybrid system of pounds, shillings and pence, the disadvantage of which is precisely that it is hybrid, and therefore does not lend itself, as the decimal system does, to a “place” and “point” system of numeration. (A suggestion for rectifying this defect will be given later in this essay.)

With all this, however, pounds and pence have an advantage which the franc and centime, dollar and cent, metre and centimetre, cannot possibly claim, namely the exceptional divisibility of the number 240. This in fact is one of those integers which mathematicians, in that special field called the “theory of numbers”, are accustomed to call “abundant”.

An abundant number is one that has more factors than any number less than it; other examples of small size are 12, 24, 36, 60,120, 360. The gross, 144, or twelve dozen, just misses abundancy, being excelled by 120. Compared with 120 and 144, even with 60, the number 100 is relatively poverty-stricken in this respect — which indeed is why the metric system is a notably inferior one; it cannot even express exactly for example the division of the unit, of currency, metrical or whatever, by so simple, ubiquitous and constantly useful a number as three.

We are therefore entitled to ask: why, in this age of scientific progress, do we endure a system of numeration with so many disadvantages?

The answer removes us at once to remote history and probably prehistory; men counted on their fingers, and to this alone, reinforced, it is to be feared, by the indolent, unreflecting, and often arithmetically illiterate force of habit, the survival of the decimal system is due.

This cannot however last; men will not always evade decision by the facile and procrastinatory cliché of our times, “not practicable in the foreseeable future”.

In later paragraphs it will be indicated how new kinds of electronic computers, and the new type of education that this will enforce in the schools, universities and colleges of technology, are bound to produce a full acquaintance with four systems of numeration at least:

(i) the binary, based on two, the foundation of all electronic computation, to the exclusion (meanwhile) of the decimal except at the final stage of conversion and recording results;

(ii) the octonary, the system based on eight, by which binary results may by the simplest of transformations be compressed and held in store;

(iii) the decimal, since unfortunately, with all its defects, it is still with us;

(iv) the duodecimal, which in the opinion of many such as the writer will prove to be that system which translates the binary to the world at large, the world of men and women behind counters, ticket offices, carpenters’ benches, in stores, in homes.

History of Numeration

With such various introductory remarks, let us look at the history of numeration. We know of course, arithmetic in primitive times being necessarily primitive, that counting and barter were done on the fingers (whence the name digit for a number-sign), and that these hardened into written marks or into such movable objects as the beads or counters on the Chinese, Japanese or Russian abacus.

On the abacus, for example, the several parallel rods carrying counters are all crossed at right angles halfway along by a fixed dividing bar; each rod has on one side of the bar five counters, on the other side a single counter. (The number five, it is interesting to note, can be represented in two different ways; either, with the thumb, push all five counters up against the bar, or leave them alone and with the finger pull that other counter back against the bar.) The abacus, used by an expert, has remarkable resource and speed; during the American occupation of Japan, a Japanese with an abacus beat an American using a hand-operated calculating machine.

The whole point of mentioning this here is that if, for example, Russia should ever go duodecimal, a not unlikely possibility which would give her people, in all the ordinary calculations of life, an advantage of at least 35 manhours—so I reckon—in every 100, China could align herself with Russia even more simply, by having six counters instead of five on the half-rod of every abacus.

Ancient History

But to return to ancient history. The Sumerians of two thousand B.C., as is shown by certain cuneiform inscriptions brought to light not so long ago, used the ten system but also the sixty, the sexagesimal system; we have for example their multiplication tables. By 1800- 1700 B.C. something quite extraordinary takes place; the Babylonians take over from the Sumerians, and while still in the market place the scale of ten persists, the astronomers, architects, in fact what one may call the mathematicians, scientists, technologists of that remote period, the Hammurabi dynasty of 3700 years ago, constitute a hierarchy skilled in arithmetic to a degree unrivalled in the modern world; for they actually used the scale of 60, the sexagesimal scale, for fractions, reciprocals, even square roots. They have left the trace of their system in the 60-fold division of the hour into minutes and the minutes into seconds, a predominantly duodecimal subdivision, as one may see by looking at a clock, but in this we observe an accommodation not so much with the scale of ten as with 5.

Another such trace is the division of the whole circumference of the circle into 360 degrees. At the time of the French Revolution certain fanatical decimalists (following in the footsteps of Stevinus of Bruges two hundred years earlier) were for dividing the right angle into 100 degrees called “grades”, the half day into ten hours, even the year into ten months. These efforts, or rather the second of them, met with no success. Astronomers and surveyors will never use so defective a system; and numbers of instances can be cited, from trigonometry, periodic analysis, approximate evaluation of areas and volumes, and so on, in which a five-fold or ten-fold subdivision of the range gives formulae and methods remarkably inferior to a six-fold or twelve-fold one.

Those Babylonian mathematicians, by the way, have extensive tables, not only of reciprocals and square roots but actually of triads of integers making the sides of a right-angled triangle, the theorem of Pythagoras 1150 years before Pythagoras; but all in sexagesimal. The central point in all this is that 60 is an “abundant” number. That was why the Babylonians, masters of arithmetic in a way that, with certain exceptions, we are not, used it as a suitable base for their numerical system.

The Egyptians were not good at arithmetic; they could “ do sums”, but even the addition of vulgar fractions was carried out by them in an unbelievably cumbrous manner.

The Greek system of numeration was an inconvenient one, letters of the alphabet being used for numbers.

The Roman was hardly better, except that with a special kind of abacus they used a duodecimal notation for fractions, traces of which survive in two of our nouns, ounce and quincunx, that is to say, a twelfth and five-twelfths. For integers, however, they used the ten system and their well-known numerals; beautiful (none better, said Eric Gill) for lapidary inscriptions and coins, of no use for convenient calculation.

These endured in arithmetic almost up to A.D. 1500, simply because of the all-pervading dominance of the Roman Empire, and later of Rome itself.

In Asia this was not so; Hindu arithmetic had evolved special single symbols for the integers up to nine, together with the zero, long believed to be a Hindu invention until lately rediscovered, in an analogous role, in Babylonian cuneiform.

This Hindu system, with its excellent “place,’ convention, though not yet extended to fractional use with the “point”, percolated to Europe by way of the Arabs (for what we call Arabic numerals ought more justly to he called Hindu-Arabic), and the geography, early steps and manner of this percolation are worth a brief interlude.

Here it is convenient for speed to link in sequence a few sentences from Cajori’s History of Mathematics: “. . . at the beginning of the thirteenth century the talent and activity of one man was sufficient to assign to the mathematical science a home in Italy.... This man, Leonardo of Pisa, ... also called Fibonacci, . . was a layman who found time for scientific study.

His father, secretary at one of the numerous factories on the south and east coast of the Mediterranean erected by the enterprising merchants of Pisa, made Leonardo, when a boy, learn the use of the abacus. During extensive travels in Egypt, Syria, Greece and Sicily ... of all methods of calculation he found the Hindu to be unquestionably the best. Returning to Pisa he published, in 1202, his great work, the Liber Abaci, . . . the first great mathematician to advocate the adoption of the ‘Arabic notation’ “.

And later we read: “In 1299, nearly 100 years after the publication of Leonardo’s Liber Abaci, the Florentine merchants were forbidden the use of the Arabic numeral(s) in book-keeping, and ordered to employ the Roman numerals or to write the numeral adjectives out in full.” The interesting parallel, but in the opposite direction of legal enforcement of innovation, is that in 1801 and again in 1837 the French introduced legal penalties against those recalcitrants who still held out against the metric system.

Arabic Numerals

The system of Arabic numerals (really, as we have just seen, Hindu-Arabic) with its “place” convention — and this, not the choice of ten at all, is the real novelty and the real advantage — was thus introduced into Europe by one man, and had to fight its way for acceptance long years after he was dead. Thus a gravestone in Baden in 1371 and another in Ulm in 1388 are the first to show Arabic and not Roman numerals. Coins are more indicative: Swiss of 1424, Austrian 1484, French 1485, German 1489, Scots 1539, English 1551. The earliest calendar with Arabic figures is of date 1518. So our authority sets down; but he may be out in slight respects.

Napier and the Decimal Point

It would be tedious for the present purpose, however interesting for leisurely investigation, to pursue this. Enough to say that the first to invent the “decimal point”, written by him as a comma, was John Napier of Merchiston, in his Rabdologia of I6I7, the year of his death and three years after the publication of his logarithms.

Then in that era following the Renaissance, mathematics and arithmetic began to make the cumulative and ever-accelerating progress which we know; and so we come, by some drastic telescoping, to where this essay began, at the years 1790, 1795, 1799, the introduction of the metric system and the decimal system of currency, which Britain, having delayed so long with instinctive, characteristic and well-founded hesitation, is now considering.

I propose to vindicate in the ensuing paragraphs the soundness of that instinct, to show that Britain need adopt nothing whatever from France, America or the apparently progressive though in fact mathematically reactionary change of system in South Africa, and to try in some measure to forecast the future of computation.

The Duodecimal System

The episode of Leonardo Pisano is significant. The supersession of Roman numerals by Arabic digits, and eventually, but not all at once, by the “place” and “pointshifting” system, was in its initial stage the work of one man of perception but above all of conviction and energy.

This strength of conviction, but now in a new and even more progressive direction, namely that the system of Leonardo is not the final word but that the duodecimal system with appropriate notation is appreciably superior again, is held at the present time by a relatively small number of persons in the whole world. (It is true, of course, that the vast majority of the rest are entirely ignorant of the whole issue.)

One may mention the Duodecimal Society of America, counting in its membership distinguished actuaries and other prominent men— and it is symptomatic that such a society should take its origin in a country devoted since 1786 (a date in which America had no mathematical standing whatever) to decimal currency, though not, and this is again symptomatic, to decimal metric; there is a Duodecimal Society of Great Britain, recently founded, small in membership and resources; while in France, home of the decimal-metric system, there is M. Jean Essig, Inspecteur-Général des Finances, whose notable treatise on duodecimal arithmetic and measures, Douze: notre dix futur (Dunod I 955), is taken seriously, as the foreword shows, by Membres de l’Institut in France and Belgium. This small band of convinced men increases its numbers all the time and gains successes here and there, as when, for example, the most recent and progressive American school-texts on arithmetic and algebra, at the secondary stage, devote an extensive chapter to the description and appraisement of “scales of notation”, leaving the pupil in no doubt regarding the relative inferiority of the decimal system.

Yet anyone who enters into public discussion on duodecimal calculation comes at once upon the strangest circumstance. Incredible numbers of persons have been so imperfectly educated as to suppose that the decimal system is the only one that admits “place” notation and the property of shifting the “point” under multiplication or division by the base.

This defect of education, amounting in the case of certain newspaper correspondents to arithmetical illiteracy, has to be combated. The fact is that any integer whatever, suitable or unsuitable, can be taken as base of the corresponding system. A younger generation of persons selected by ability knows this already, namely all those who are preparing themselves for modern electronic computation, destined as it is, in the form of new machines not yet in production but easily imaginable, to transform in a hardly recognisable way whole domains of financial and official calculation, to say nothing of the arithmetical apparatus of technology generally.

For while1900-1925 was the period of the hand-operated mechanical calculating machine, and 1925 and onward that of the electrical one, from 1961 to the end of the millennium will be the era of electronic computers of every range, not merely of the large, and for certain purposes too large, ones that we see being installed in more and more places, but those of moderate size (and there will be smaller ones still) which are only now beginning to be in production. These will transform not merely arithmetic, but education in arithmetic; and a younger generation, familiar with binary and octonary systems as well as with decimal, will be sure to ask: What, reckoned in terms of time and efficiency, is the worth of the decimal system, and is there a better?

We shall without doubt see this happen, probably in Russia and America almost simultaneously, while we, who of all nations in the world are in the special and most favourable position to make the change, may be left behind; may well in fact have made a belated change, only to have to make a further belated one. Of course, on the other hand, there may be financial, economic and indeed political considerations which may enforce the other, to my mind reactionary, decision; but that would require a separate study, which has in some part been done and is in any case outside my competence. But I will simply say: political expediency is the ruin of science.

Monetary and Metrical Units

Why are we in that special and most favourable position? Because we already have the duodecimal system with us in all but name, and to a certain but lesser extent even in notation. I refer not to electronic machines, which can convert from their idiomatic binary into any other prescribed scale, but to the numberless transactions of ordinary life, in banks, ticket offices, behind counters, on board buses, wherever and whenever there is buying and selling and giving of change. Consider a railway clerk giving tickets and change, often at top speed to a heavy queue.

Does he ever think of decimal tables in handing back 5s. 7d. as change from a 10s. note on a ticket of 4s. 5d.? Not he; like hundreds of thousands of men behind counters he is a highly versed duodecimalist, though it would not occur to him to give so publicly useful a faculty so highsounding a name. I know this from having spoken recently with dozens of such men.

Here is a typical comment, from a Scots bus conductor: “We get on weel eneuch; yon would muck it all up again”. Some may think they might get on weel eneuch with decimal coinage; the most manage perfectly well. There is no cogent evidence that the public wish this change in the least; though the will of the public, strong as it might be either way, is neither the only nor the chief consideration. The French, at the very height, in 1790, of their enthusiasm for liberty, equality and fraternity, so qualified equality as to set up an academic commission of the most distinguished mathematicians in the land.

However, I propose — and it is not at all original with me — a certain change, a slight one, by which in a phased gradualness, an interregnum of years of quiet habituation and consolidation, we may bring in the more efficient system. It is: to have a pound, call it R for this discussion (a stag of twelve points is a royal!), of twelve shillings, a gross of pence.

It banishes at a stroke all oddments from twelve shillings and a halfpenny to nineteen shillings and elevenpence halfpenny; it is a paper note, a “royal”, that mediates between and supersedes the pound and ten-shilling note, requires no new minted coinage whatever, and is very close to one and two-thirds dollars. Call it then R1:0:0. Its half is R0:6:0 shillings. Its quarter R0:3:0. Its eighth R0:1:6. All very much as at present.

The half-crown might stay for a while, but eventually might be superseded by a three-shilling piece, a “quarter”, easier than the halfcrown to distinguish from the florin. Pennies and the rest are exactly the same as now. For example, except that we have this R of new value, we shall write R3:0:0; and the like as before.

So also for feet and inches. There might be — I do not know whether it is suitable or not, and would not presume to dictate to the practical measurer — a new “rod” simply of twelve feet, and this would make parallelism complete.

Duodecimalists should not dictate too much what is desirable; they may well leave it to practical craftsmen to find what is the best accommodation, provided only that the final outcome is indeed cast in a duodecimal hierarchy of units.

Here I differ from many duodecimalists; for I believe that, if the principle is once accepted, practical and intelligent men can be trusted to find possibly an even better solution than any duodecimalist or duodecimalist society might have proposed.

General Arithmetic However, to go further, let us pass from the monetary or metrical units and super- or subunits to the general arithmetic of the matter. Thus, let the fraction a half itself, in whatever context, be denoted by 0:6, a third by 0:4, a quarter 0:3, a sixth on:, twelfth 0:1, where the colon (most duodecimal publications use a semi-colon) serves for the duodecimal point, and will move right or left under multiplication or division by twelve.

For example, movement to the left. What is a twenty-fourth? A twelfth of a half, hence 0:06; a thirty-sixth is 0:04. A thirtysixth of the new royal is indeed fourpence; and so on. Contrast this with the inexact and inadequate third as 0. 33333 . . ., sixth as 0.6666 . . ., twelfth as 0.083333 . . ., and so on to more turgid examples.

Someone may say: What about a fifth or a tenth? Admittedly, since five does not go exactly into twelve, we shall here obtain a non-terminating duodecimal. For example, a tenth comes out as 0:12497 . . .. the last four digits forming the recurring period; but a close approximation to this is 0:125, committing the slight error, in excess, of 1/8640. (For comparison the approximation 0. 333 for one-third commits, in defect, an error of one three-thousandth.)

However, to go slightly further still. A shilling, 1:0s., is a dozen pence. Shift the colon to the right and in fact, since it is not then necessary, remove it, and write the dozen itself as *10, the prefixed asterisk (functioning like the American dollar sign) indicating that we are in a special system, that of the dozens, the meaning of the symbols being: one dozen, no units. Similarly thirteen, being one dozen, one unit, is *11; fourteen is *12, twenty-five is *21, and so on. The gross likewise is *100, meaning one gross, no dozens, no units; I will attend to names later. But all of this is just another way of writing 1:0:0 in the new R way, the kind of thing that faces us every day on a bill. Duodecimalism is nothing but this, though of course we have to know our tables, e.g. that 7 times 9 (asterisk with single-digit numbers not required) is *53, five dozens and three. But this is the smallest part, in a slightly different notation, of the first entries in any ready reckoner, and we have seen that already great sections of the population know these elementary tables, from habit, from serving customers and giving change. Consider the number, in decimal notation, 457. It is three gross, two dozen and one, *321. If these happened to be pence, then, in pounds, R3:2:1; in shillings, *32:1s., three dozen and two shillings and a penny. But this is to labour the habitual; we are doing this kind of thing all the time. Everyone who knows (some do not) that twelve articles at sevenpence each is seven shillings is simply saying that a dozen times seven is seven times a dozen, namely *10X7=*70 in pence, or in shillings *10X0:7s. = 7s. I showed some of this, doing some simple addition of fractions by it, to a bank teller and likewise to a stationer. The reaction was identical; each man involuntarily shielded his eye with his hand, doubtless to ward off the blinding flash of the obvious. Well, it is that some of this, in a differently couched and very uninspiring form, is taught in the chapter of school algebra dealing with “scales of notation”, though often treated in such a perfunctory fashion that the pupil may be excused from regarding it, as so much tediously useless manipulation. I exclude from my condemnation those admirable American school textbooks. We have suggested, provisionally, *10 for twelve, *11 for thirteen: for we hope eventually to use our system exclusively and to drop the asterisk. Confusion will be caused unless we devise new single symbols for ten and eleven; we can keep the names. Is it beyond the power of artistic typography (I suggest 7, an inverted 2, for ten; and 8, an inverted 3, for eleven) to invent simple, distinctive, cursive and aesthetically satisfying symbols for these two integers? The Hindus had to invent all ten of their symbols; while I could show many unsuspected situations in ordinary arithmetic where an alternative ten, at least, would have been valuable. On the Chinese and Japanese abacus there were and are two ways of expressing five, appropriate to different situations. For myself, I do my calculations with no great need for symbolic representation, but the above inversions of 2 and 3 served me well enough. Certain duodecimal societies, as well as a good many idiosyncratic individuals, have advocated various symbols, quite commonly X or c for ten, E for eleven, and so on. This will not do: letters of the alphabet must be kept for algebra, not arithmetic; let us think of the confusion in trying to write in such a way “ten times X”. So also for nomenclature.

For myself, I do not depend much on auditory impression for number, but thinking of the Scots”twal” I sometimes imagined “twel-one”, “twel-two”, and so on for thirteen, fourteen and the rest; but of course in dictation one would mention “asterisk” and call out, just as we do in decimal, “one one”, “one two” and the like for *12, *12, etc. There should be no difficulty here. Once again, duodecimalists should not prescribe too much for others in this matter; language and linguists should be able to find, as the French language does with never failing felicity, euphonious and idiomatic equivalents for any new entity that may arise. For example Icelandic also, when faced with the

**necessity of finding words for radio, television and so on, merely drew on its own resources. Let the principle be once stated; we can weigh later the merits of the different suggestions.**

As for early education in the properties of numbers, it is evident that twelve is a far more interesting number than ten, and two sets of six or twelve coloured blocks, to be arranged in various ways by twos, threes, fours and so on, would show to the growing mind the mutual relations of small integers better than any of the usual devices based on ten, some of them in any case open to criticism. Above all, no dependence on fingers. This will be enough of description for a first summary. A graduated set of simple exercises would lead anyone, even a child, easily into this realm thus simplified. But it will be asked: are the reasons for change aufficient, both qualitatively and quantitatively, to justify, so late in the history of the world, such a radical transformation of mental habit and customary practice? The replies are: First, it is very early in the history of the world. Second, that in our case at least, the change is not radical; we do much of it already every day. Third, partly qualitative, that since the dozen, helped by its multiples and submultiples, is so extraordinarily superior to ten in all that concerns parcelling, packaging, arrangement, subdivision, to say nothing of a host of applications which could be cited from mathematics, the practical use of the dozen and its adjuncts should go hand in hand and step for step with the corresponding numerical use; and this implies the duodecimal system and no other.

Finally, the quantitative advantage. To begin with, the multiplication tables are simpler than the decimal ones; there are only 55 (duodecimally *47) essential products to be learned, exactly the same number as have to be learned in our school tables up to twelve times twelve—and observe that even there we had to go to the dozen. (Incidentally in duodecimal the square of *11 is *121, of *12 is *144, with different numerical meaning, of course.) For multiples of 2, 3, 4, 6, 8, 9 and 10 we see in the last digits a simple and useful periodicity. For example, the four times table: last digits 0, 4, 8, 0, 4, 8, 0, 4, 8, and so on; the three times table: last digits 0, 3, 6, 9, 0, 3, 6, 9 and so on. Tests for divisibility: for divisibility by 2, 3, 4, 6, look at the last digit only; by 9, 16, 18, the last two; and so on.

Duodecimal fractions, as we indicated by a few examples earlier, are in the usual fundamental ones of low denominator remarkably simpler than decimal. Consider the table below:

Fraction Decimal Duodecimal

1/2 0.5 0:6

1/3 0.3333 0:4

1/4 0.25 0:3

1/5 0.2 0.2497

1/6 0.1666 0:2

1/8 0.125 0:16

1/12 0.0833 0:1

1/24 0.04166 0:06

Tables of successive halvings, as for example the table for conversion of sixty-fourths into decimals that hangs on the wall of many tool shops, shows comparisons such as the following five: thus

Fraction Decimal Duodecimal

25/64 0.483

27/64 0.421875 0:509

29/64 0.453125 0:553

31/64 0.484375 0:599

33/64 0.515625 0:623

With only three digits, the duodecimal fractions are all exact. Comment is needless.

But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal.

Others (but so far I have not heard of even one such investigator) might arrive at a slightly different estimate; but I am certain that in every case a marked superiority for the duodecimal system would be established. If such a waste of time and effort (about 350 hours lost in every 1000) were found to be trickling away in any department of a modern production unit, a time-and-work study would at once be set up. Some altruist might even come in with a take-over bid. Is it to be doubted that such time, saved and turned to more productive ends, social or economic, would give an advantage much outweighing any advantage assumed to accrue now, at this late stage of decision, from moving over to the decimal system; an assumption moreover implying, since the decision has taken about 150 years to make, that the new status of things would last for at least another 150 years.

Nothing stands still, not even arithmetic. That arbitrary division of time, the second millennium, is approaching, heralded as it has been somewhat prematurely from a distance of forty years; and no doubt a few thousands of superstitious decimalists will sit up on that eve to await the new dawning of heaven and earth. In the interim there is bound to be incredible technological progress, enough possibly to give us some glimpse of “the uses of leisure”. Among these novelties the transition from a defective system of numeration and metric, to a new one, attainable by easy and gradual phase, will be viewed in remote retrospect as one of the most ordinary pieces of belated tidying-up that ever was delayed for so long past its due time. It will be viewed, indeed, by the future historians of mathematics, as completing the work of Leonardo, in a direction which, with the added knowledge of 800 years, he would have approved.

(First published 1962)

As for early education in the properties of numbers, it is evident that twelve is a far more interesting number than ten, and two sets of six or twelve coloured blocks, to be arranged in various ways by twos, threes, fours and so on, would show to the growing mind the mutual relations of small integers better than any of the usual devices based on ten, some of them in any case open to criticism. Above all, no dependence on fingers. This will be enough of description for a first summary. A graduated set of simple exercises would lead anyone, even a child, easily into this realm thus simplified. But it will be asked: are the reasons for change aufficient, both qualitatively and quantitatively, to justify, so late in the history of the world, such a radical transformation of mental habit and customary practice? The replies are: First, it is very early in the history of the world. Second, that in our case at least, the change is not radical; we do much of it already every day. Third, partly qualitative, that since the dozen, helped by its multiples and submultiples, is so extraordinarily superior to ten in all that concerns parcelling, packaging, arrangement, subdivision, to say nothing of a host of applications which could be cited from mathematics, the practical use of the dozen and its adjuncts should go hand in hand and step for step with the corresponding numerical use; and this implies the duodecimal system and no other.

Finally, the quantitative advantage. To begin with, the multiplication tables are simpler than the decimal ones; there are only 55 (duodecimally *47) essential products to be learned, exactly the same number as have to be learned in our school tables up to twelve times twelve—and observe that even there we had to go to the dozen. (Incidentally in duodecimal the square of *11 is *121, of *12 is *144, with different numerical meaning, of course.) For multiples of 2, 3, 4, 6, 8, 9 and 10 we see in the last digits a simple and useful periodicity. For example, the four times table: last digits 0, 4, 8, 0, 4, 8, 0, 4, 8, and so on; the three times table: last digits 0, 3, 6, 9, 0, 3, 6, 9 and so on. Tests for divisibility: for divisibility by 2, 3, 4, 6, look at the last digit only; by 9, 16, 18, the last two; and so on.

Duodecimal fractions, as we indicated by a few examples earlier, are in the usual fundamental ones of low denominator remarkably simpler than decimal. Consider the table below:

Fraction Decimal Duodecimal

1/2 0.5 0:6

1/3 0.3333 0:4

1/4 0.25 0:3

1/5 0.2 0.2497

1/6 0.1666 0:2

1/8 0.125 0:16

1/12 0.0833 0:1

1/24 0.04166 0:06

Tables of successive halvings, as for example the table for conversion of sixty-fourths into decimals that hangs on the wall of many tool shops, shows comparisons such as the following five: thus

Fraction Decimal Duodecimal

25/64 0.483

27/64 0.421875 0:509

29/64 0.453125 0:553

31/64 0.484375 0:599

33/64 0.515625 0:623

With only three digits, the duodecimal fractions are all exact. Comment is needless.

But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal.

Others (but so far I have not heard of even one such investigator) might arrive at a slightly different estimate; but I am certain that in every case a marked superiority for the duodecimal system would be established. If such a waste of time and effort (about 350 hours lost in every 1000) were found to be trickling away in any department of a modern production unit, a time-and-work study would at once be set up. Some altruist might even come in with a take-over bid. Is it to be doubted that such time, saved and turned to more productive ends, social or economic, would give an advantage much outweighing any advantage assumed to accrue now, at this late stage of decision, from moving over to the decimal system; an assumption moreover implying, since the decision has taken about 150 years to make, that the new status of things would last for at least another 150 years.

Nothing stands still, not even arithmetic. That arbitrary division of time, the second millennium, is approaching, heralded as it has been somewhat prematurely from a distance of forty years; and no doubt a few thousands of superstitious decimalists will sit up on that eve to await the new dawning of heaven and earth. In the interim there is bound to be incredible technological progress, enough possibly to give us some glimpse of “the uses of leisure”. Among these novelties the transition from a defective system of numeration and metric, to a new one, attainable by easy and gradual phase, will be viewed in remote retrospect as one of the most ordinary pieces of belated tidying-up that ever was delayed for so long past its due time. It will be viewed, indeed, by the future historians of mathematics, as completing the work of Leonardo, in a direction which, with the added knowledge of 800 years, he would have approved.

(First published 1962)