**Words Are As Seeds To The Wise - But Are As Claws To The Shallow**

__Latest Updates__**January 20: Religion and Science page re-titled as Jesus Mary And Judas The Foulest Lies Of All, new chapters and the order of chapters reformatted.**

**November 8:**

**Climate Change Page the first section of the revised page Emailed to...**

**IPCC-Sec@wmo.int info@theclimategroup.org theecologist@theecologist.com info@carthagegroup.com press@carthagegroup.com**

office@vic.greens.org.au office@act.greens.org.au office@nsw.greens.org.au office@qld.greens.org.au office@wa.greens.org.au office@tas.greens.org.au office@sa.greens.org.au office@vic.greens.org.au

__Recommended sources of actual and factual information.__**International News and Weather Al Jazeera.**

The New Rulers Of The World by John Pilger.

A Secret Country by John Pilger.

The New Rulers Of The World by John Pilger.

A Secret Country by John Pilger.

**Really Graceful Documentaries - Educational Videos on You Tube ***Highly Recommended*****

****

A NATURAL CIRCLE

A 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.**

__Simply__**THREE TIMES THE LENGTH OF A LINE IS THE LENGTH OF ITS CIRCLE**

CONCERNING THIS HOMEPAGE

CONCERNING THIS HOMEPAGE

**Some three years ago I started this website as a means of transferring and presenting work taken from my old website www.geometry-mass-space-time.com in a revised and more organised manner.**

**The old website to say the least is very disorganised, and the reason for this is because I did not become involved with computers until some twenty years ago. And this, combined with the analogue**

*(empirical-practical-pragmatic)*nature of my mind, as opposed to decimal/digital computing, precluded any interest in my using a computer for any other purpose than to type out my work, and then store it for later publication.**Over the years, as the number of interrelated subjects and work involved increased, and noting that personal websites became more available, I decided that a website might be ideal for reorganising and storing my work, while also being able to publish it directly online.**

**However, as the thought processes of the sub-conscience mind are branching in nature, and as many of you will be aware, it chooses its own time as when to pop into your head, and then in effect it**

**forces you to think about a matter or a subject regardless as to whether you want to or not.**

**Mind map - Wikipediahttps://en.wikipedia.org/wiki/Mind_map**

**So it was, given the increasing number of subjects, and the amounts of mental and textual work involved in covering and explaining them, and this, combined with my computer illiteracy, it was inevitable that my first website would become a disorganised mess.**

**As such, and realising that it would be time consuming, and very difficult to reorganise the website into a semblance of good order, and at the same time my being introduced to WEEBLY. I decided to relegate the first website to being archival and for reference purposes only, and start a new website,**

**whereby I could devote one page to a particular subject matter, and also add illustrations and images as needed**

**The archived work on my other website remains in its**

**original state, and still contains errors that I made during the early days of my considerations. However, I have chosen to leave them in place, because, although they served to lead me along many blind alleys before realising my errors. They also taught me, that it is only by seeking to recognise and then admit to our mistakes, that we are able to progress on along a path leading to truth.**

**By leaving them in place, and as they relate to the corrected, revised and more advanced thinking on this website. I hope they will serve others to realise that we should not make assumptions or accept the first thing we are told, or comes into our head, without first taking time to think/question the matter through, and then continue to rethink it through. Until we finally manage to rule out all of the blind alleys, and find the right question to ask ourselves**

*(Our: Subconscious Mind - Depressed Mind - Meditative Mind - Memmorative Mind - Analogue Mind - Analogue Hard-drive - Id - Soul - Super Ego - Me - I*), and the correct answer finally pops into our heads and then it makes sense.**And It is only in this way, by recognising, accepting and admitting to our mistakes, and taking note of the details and storing them away for future reference, that we are the able to be clear of them, and then move on to build a mind-map of truth; that is inviolable to the taught and enforced coercive beliefs, opinions, lies and the cons of others.**

**E.g.**

The persuasive claims of parasitic work-shy and dishonest money-lenders - corporate owned lawyers - politicians - marketers - advertisers - retailers and other shysters.

The persuasive claims of parasitic work-shy Shamans - Witch-doctors - Priests -

The inane non-empirically based or rationally proven theories of parasitic work-shy Science

The persuasive claims of parasitic work-shy and dishonest money-lenders - corporate owned lawyers - politicians - marketers - advertisers - retailers and other shysters.

The persuasive claims of parasitic work-shy Shamans - Witch-doctors - Priests -

__Theo__logians and other self- righteous hypocrites and shysters.The inane non-empirically based or rationally proven theories of parasitic work-shy Science

__Theo__rists.

**DIFFERENTIAL GEOMETRY**

(Curved Geometry)

(Curved Geometry)

**SIMPLIFIED FOR**

**EVERYONE**

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

**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.

Using a 120-centimeter 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.

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

For average

__Rhetorical Questions__For average

**laypersons 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 to the area of the outline of a circle, as opposed to using 3 times the circles diameter to find the exact length of a circle, which would you choose to use?**

**For educators and those of the**

**professions.**

Given the number of Pi

Given the number of Pi

**3.14285714285**

**calculations which are carried out each day among a world population of more than seven billion people.**

**H**

**ow many trillions of irreplaceable and accumulative hours/days/years of human lifetimes could be saved and put to a better and more constructive use, over the space of a single year of 365 days?**

Simply by doing the right thing and using the 3 to 1 ratio of a circle to find its exact length, rather than the approximation - abomination of a circle Pi.

Simply by doing the right thing and using the 3 to 1 ratio of a circle to find its exact length, rather than the approximation - abomination of a circle Pi.

**Open challenge to any mathematician, geometer, or so-called theoretical mathematician or physicist**

*(*__and__

__anyone____else__

*to disprove or discredit the simple arithmetic of the circle as provided previously or the remainder of simple arithmetic which follows on this page.*__worldwide__)

**FOLLOWING CONTENT**
Editor In Chief ATHENA PRESS Publishers, Readers Report 2005Introduction Drawn Lines Pi Sumeria 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.Tw elve Steps From A Cube To Its Sphere.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 YouTube Comments: Reality Versus Fiction |

INTRODUCTION

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 Euclid and Archimedes, to this very day, not one of any of the millions of**

**Geometer's worldwide is able to**

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

__DRAWN LINES__**A drawn line, is a visually apparent length of distance,**

Or to

__that has been artificially made apparent,__by applying an__overlaying and contrasting____"linear - area"__of shade, colour, or texture, to the distance between any two given points on the drawing surface.Or to

__"out - line",__and so__"____enclose an amount of surface area",__in the form of a shape, by applying an__overlaying and contrasting____"linear -____area"__of shade, colour, or texture, to sub-divide the surface area, into__"three distinct parts__of the drawing surface.**Questions**

**When we look at the shape of a bright yellow full moon as it is being silhouetted against the dark background of the night sky, does the full moon have a circumference – circumferential outline?**

**Answer**

**No, it does not; the full moon is a yellow coloured round circular area of shape; which is being contrasted against the greater surrounding area, of the darkness of the night sky. to produce a round silhouetted circular shape that does not possess an outline.**

**If we take a black marker pen and draw a black circle at the centre of a sheet of yellow A4 paper, does the yellow round circular shape in the middle of the paper have a circumference - outline?**

**Answer**

**No, it does not; the yellow round circular area of shape in the middle of the paper is being contrasted against the surrounding area of blackness belonging to the circumferential thickness of another circumventing black circular shape. And the circumferential thickness of the area to the black circular shape is in its turn is being contrasted against the lighter background of the rest of the yellow A4 paper.**

**Questions**

**When we look at a tree in the brightness of day light, does the shape of the tree possess an outline?****When we look at a full Moon at night, does the circle of the Moon possess an outlines**

**Answers**

**No, they do not, the darker area belonging to the shape of the tree is contrasted against the greater surrounding area, of the brightness and blueness of the sky.**

When we look at the circle of a full Moon at night, the bright yellow of the circular Moon is contrasted against the greater surrounding area of the darkness of the night sky.

When we look at the circle of a full Moon at night, the bright yellow of the circular Moon is contrasted against the greater surrounding area of the darkness of the night sky.

**Simply**

**Shapes are not geometric; they are the natural visual forms of all things that exist in nature, which are made visually manifest by the presence of a contrasting background.**

**Contrasting backgrounds make natural shapes visually manifest according to six aspects of lighting and visibility; shades of darkness, shades of brightness, shades over distance, shades of perspective, shades of colour, shades of texture.**

**Pi**

**Is not an equation or a number it is a Greek letter/symbol which was given to Archimedes improper fraction of 22/7, and near to approximation to a circles actual length.**

**OXFORD E**

**nglish Dictionary**

**Pi: The symbol of the “ratio” of the “circumference” of a circle to its diameter length “approximately” 3.14159.**

**Mathematicians first began using the Greek letter π called Pi as a symbol in the 1700's. following its Introduction by William Jones in 1706, and the use of this symbol was later popularized by Leonhard Euler, who adopted it in 1737.**

**The original Archimedean *approximation* of a circles length was given as the improper fraction of 22/7 (3 whole units and 1/7th).**

L

L

**ater at some point in history the approximation of 22/7 was converted into a decimal by subdividing the number 22 above the line by the 7 below the line.**

**This yielded the irrational figure of 3.14285714285 which equates to three whole units of diameter length, with .14285714285 smaller parts of 1/7th of a unit of diameter length; continuously subdividing down into a never ending infinite stream of smaller parts of the 1/7th of diameter length.**

**The improper faction of 22/7 when it is properly cancelled out using common arithmetic and dividing the lower 7 into the 22 parts above the line equates to three diameter lengths, with each diameter length measuring seven measurement units, and 1/7th of one diameter length left over.**

**And quite simply if we disregard all of the historically disingenuous nonsense's and complexities which have been used to surround the simple 3 to 1 circle with, we can envision what took place when this so-called formula was originally formulated by Archimedes.**

**The cylinder that Archimedes used for his calculations obviously had a diameter length consisting of seven smaller measurement units, thus giving a total of 21 smaller measurement units to its length of circle.**

**However, given that the only way that Archimedes could physically measure the length of the cylinders circle,**

**was by use of a flexible piece of string.**

**As Per Diagram**

**So it was, that when he came to straighten and measure his piece of string against a straight rule, he failed to take into account the thickness of the string, and so failed to understand that he was measuring the outer stretched length of the string, rather than the shorter inner constricted length of the string, which had been been directly in contact with the cylinders circle.**

**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).**

**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.**

**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 Arithmetic of Jewish-Grecian-Roman Capitalism

THE

Fraudulent

Decimal Arithmetic of Jewish-Grecian-Roman 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.

**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)