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Content

Content

Sumeria 1000 BCEuclidean Straight Three - Dimensional Geometry The length of a Circles EdgeThe Sumerian Method For Calculating The Area Of A Circle The Square Of A Diameter Is The Square Of The Circle Archimedes Proposition 1 - Archimedes TriangleThree Time The Radius Squared Four Quadrants Squaring The Circle Calculating Areas Of RingsCalculating From A Cube To Its Sphere Twelve Steps From Cube To Sphere The Areas And The Volumes Of Symmetrical Ovals And Ovoids The Number Of Degrees Of Latitude And Longitude To A Sphere The Area Of One Degree Of The Surface of A Sphere |
The Irrationality Of π Circle - Versus Circumference Euclidean Definitions Negated The Twelve Square Phenomenon Of The Pythagoras Theorem Reference IFL Science: Ancient Babylonians Used Geometry To Track Jupiter Thousands Of Years AgoThe Case Against Decimalisation by Professor A C Aitken (Formerly Professor of Mathematics, University of Edinburgh) e.mail:universometry@gmail. unialphaomega@hotmail.com© www.fromthecircletothesphere.net ©www.geometry-mass-space-time.com You-tube "Reality Versus Fiction" Comments sections of videos, as per subjects covered here on this website. |

**SUMERIA 1000 BC**

**

Reference: Estimating the wealth; Encyclopedia Britannica.

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***

**The sheer lack of insight conveyed in this comment is truly astounding.**

**The Sumerians were masters of mathematics, straight linear geometry, and curved differential geometry millennia before the Greeks came along, and it was they who using their skills in differential geometry, were the inventors of clocks and time.**

**Fundamentally**

**A circle is a round shape, whose curvature consists of a continuing series of points equidistant from a fixed point (its center).**

**The curved edge length of a round shape is a continuing series of points equidistant from a fixed point (center).**

**The circumference of a circle is the width (area) of a "radiated" drawn line that serves to surround and enclose the radiated shape (area) of the circle.**

**Oxford English Dictionary**

**Circle: A round plane figure whose boundary (the circumference).....consists of points equidistant from a fixed point (center).**

**The first part of this dictionary definition is based in/on the Euclidean linear thinking of more than two millennia ago.**

**The second part of this dictionary definition is more on par with modern day physics, whereby it can be understood that the equidistant points to the circle’s center are spatial points, not linear.**

**Circles' shapes and bodies do not have any boundary lines as per Euclidean thinking; all circles have a quantity of area. And all bodies have a quantity of volume.**

**When we look at, e.g a dinner plate from a Euclidean perspective, we see it outlined against its background, just as we observe a silhouette being outlined against a lighted background.**

**Whereas, when we look at a dinner plate from a physicist's perspective, we are aware that there is no outline as such, but rather, that it is the denser atomic structure and the reflecting color of the plate which is simply standing out from the far less dense, colorless, invisible and surrounding atmosphere.**

**And in essence this is no different to what happens when we look out into the night sky to observe the linear constellation patterns of the stars, patterns which in reality do not exist anywhere except in our minds eye, and this is only because we have either been taught them or we have learned of them.**

**EUCLIDEAN**

STRAIGHT THREE - DIMENSIONAL GEOMETRY

STRAIGHT THREE - DIMENSIONAL GEOMETRY

******FOUR CURVED ORBITS - IN FORWARD MOTION**

THROUGH

DIRECTIONLESS SPACE

THROUGH

DIRECTIONLESS SPACE

EUCLIDEAN DEFINITIONS

EUCLIDEAN DEFINITIONS

__Euclid:__A point is that which has no part.**Oxford English Dictionary.**

Point: (in geometry) something having position but not spatial extent, magnitude, dimension, or direction, for example the intersection of two lines.

Point: (in geometry) something having position but not spatial extent, magnitude, dimension, or direction, for example the intersection of two lines.

____

**Rationale****If a point has no part, then it can only be due to the said point being a designated point in space distance volume or time. rather than a linear geometric point of for example the intersection of two lines.**

**Given two identical line widths intersecting (crossing) e.g. vertical and horizontal; the intersection point will have an area that is equal to the square of one line width of their cross section.**

__Euclid:__A line is a breadth-less length.

**Rationale****For a line to be breadth-less, it has to be a theoretical or an imaginary line such as e.g. the line patterns of star constellations; rather than a transcribed line which has a transcribed area of width, defined by the sharpness of a pencil's point.**

__Euclid:__The extremities of a line are points.

**Rationale****The extremities of a line are its two termini (ends) of length; any line other than the line of a differential shape e.g. a circle or an oval has two termini generally “nominated” as being A to B.**

__Euclid:__A straight line is a line which lies evenly on itself.

**Rationale****LINEAR GEOMETRY**

**A straight geometrically drawn line may be considered to be straight when drawn upon an even straight flat surface. All straight lines of flat surfaces are as with all flat surfaces, finite.**

**LINEAR PHYSICS**

**A geometrically transcribed line on an even straight flat surface, cannot be considered to be straight below the level of 20 20 vision, or under electron microscope examination.**

**All trajectories of straight line transcribed upon the Earth’s surface are curved over the Earth’s surface.**

**All straight trajectories of e.g. aircraft and missiles travelling above the Earth’s surface are gravitationally curved over the Earth’s surface.**

**The visible three-dimensions of crystalline-based bodies and things do not possess outlines, as their atoms of structure merge directly with the atoms of the surrounding atmosphere.**

__Euclid:__A surface is that which has length and breadth only.

**Rationale****LINEAR GEOMETRY**

A flat surface which has length and breadth only is a rectangle or a square.

A flat surface which has length and breadth only is a rectangle or a square.

**A flat surface possesses a finite area to its shape, and shapes possess varied linear angles and lengths, to the straight or curvaceous boundaries of their particular shape.**

**A spherical surface is not finite, as it does not have a finite length or breadth**

LINEAR PHYSICS

A surface is not flat below the level of 20 20 vision, or under electron microscope observation.

LINEAR PHYSICS

A surface is not flat below the level of 20 20 vision, or under electron microscope observation.

__Euclid:__The Extremities of a surface are**line.**

**Rationale****The extremities of a surface may be defined by the use of transcribed lines.**

**The extremities of a surface may be defined by varying types of boundary.**

**The extremities of an area of land may be limited by natural features e.g. a cliff edge, or an ocean.**

**The extremities of surfaces belonging to bodies and things do not possess any lines.**

The boundary of any shape or body, is defined by the merging of two dissimilar densities of matter.

The boundary of any shape or body, is defined by the merging of two dissimilar densities of matter.

**THE LENGTH OF A CIRCLES EDGE**

**Using a 120-centimeter length of diameter multiply this by 3.**

**The circle's edge length is 360 cm long.**

**The circle's edge has 360 degrees of subdivision.**

**The circle's edge has 360 degrees, and each degree is 1 centimeter long.**

**SUMERIAN METHOD**

CALCULATING THE AREA OF A CIRCLE

CALCULATING THE AREA OF A CIRCLE

**Using a 120-centimeter length of diameter multiply this by 3.**

The Circles Edge is 360 cm long.

Multiply the 360 centimeters "Edge Length" by itself = 129, 600 square centimeters.

Divide 129, 600 by 12 = 10, 800 Square Centimeters to the Area of the Circle.

The Circles Edge is 360 cm long.

Multiply the 360 centimeters "Edge Length" by itself = 129, 600 square centimeters.

Divide 129, 600 by 12 = 10, 800 Square Centimeters to the Area of the Circle.

THE SQUARE OF A DIAMETER

IS THE

SQUARE OF THE CIRCLE

THE SQUARE OF A DIAMETER

IS THE

SQUARE OF THE CIRCLE

**Use one Right Angle of the 120 x 120 centimeter Square as a Diameter then multiply by 3.**

**The Circle’s edge is 360 cm long.**

**Multiply the 120-centimeter diameter by 4.**

**The Square’s edge is 480 centimeters long.**

****

**ARCHIMEDES**

**287 - 212 BC**

**.**

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

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

**Archimedes Triangle**

**The circle in question has a 120-centimeter diameter length.**

**The base right-angle is equal to the radius of 60 centimeters.**

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

**The 360-centimeter height of the right-angle is equal to 6 x the 60-centimeter radius length.**

**1 radius 60 centimeters x 360 (6 radii) centimeters is 21, 600 square centimeters the area of the rectangle.**

**Half of the rectangle is 10, 800 square centimeters.**

**The area of the triangle is half of the 1r x 6r rectangle.**

**Half of the 1r x 6r rectangle is 1r x 3r.**

**(1r) 60 centimeters x (3r) 180 centimeters is 10, 800 square centimeters.**

**THREE TIMES THE RADIUS SQUARED**

**

****

The diameter of the circle is 120 centimeters.

The diameter of the circle is 120 centimeters.

**The diameter x 120 centimeters gives, 14, 400 square centimeters to the square of the diameter.**

**The 60-centimeter radius x 60 centimeters gives, 3, 600 square centimeters to the square of the radius.**

****

**The square of the radius x 3 gives, 10, 800 square centimeters to the area of the circle.**

**Sum**

**The area of the circle is 3/4 that of the square of the diameter.**

**FOUR QUADRANTS**

******Squaring a 120-centimeter diameter of a circle gives an overall square of 14, 800 square centimeters.**

**The 60-centimeter radius of the circle’s diameter, gives 3, 600 square centimeters to one quadrant of the overall square.**

**One 3, 600-square-centimeter quadrant divided by 4 is 900 square centimeters.**

**Multiplying 900 square centimeters by 3 yields 2 - 700 square centimeters.**

**Multiplying 2 -700 square centimeters by 4 yields 10 - 800 square centimeters in the circle.**

**The 10 - 800-square - centimeter area of the circle is three-quarters that of the area of the 14, 400 – square - centimeter square of the circle's diameter.**

****

**Sumerian Method 1000 BC; 10,800 square centimeters**

Archimedes Triangle 212 BC; 10,800 square centimeters

Three Times The Radius Squared 2017 AD; 10,800 square centimeters

Four Quadrants 2017 AD; 10, 800 square centimeters

Archimedes Triangle 212 BC; 10,800 square centimeters

Three Times The Radius Squared 2017 AD; 10,800 square centimeters

Four Quadrants 2017 AD; 10, 800 square centimeters

**THREE MILLENNIA**

&

FOUR IDENTICAL RESULTS IS NOT A COINCIDENCE

&

FOUR IDENTICAL RESULTS IS NOT A COINCIDENCE

ABSOLUTES

ABSOLUTES

**The length to the edge of a circle is 6 times the circle's radius and 3 times its diameter.**

**The length to the edge of a circle is three-quarters the length of the square of the circle's diameter.**

**The area to the shape a circle" is three-quarters that of the square of the circle's diameter.**

**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)***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**

**CALCULATING FROM A 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 CUBE TO SPHERE**

**CUBE TO CYLINDER**

**Calculating the surface area and volume of a 6-centimeter diameter sphere, obtained from a 6-centimeter cube, note that we are using 6 centimeters rather than 120 centimeters to make the numbers easier to follow.**

**Measure the height of the cube to obtain the length of the diameter line of 6 centimeters.**

**Multiply 6 cm x 6 cm to obtain the square area of one face of the cube, and its perimeter length.**

**24 centimeters and 36 square centimeters.**

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

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

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

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

**Divide the cube's surface area by 4, to obtain one-quarter of the cube's surface area = 54 square cm.**

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

**CYLINDER TO SPHERE**

**Divide the cylinder's cubic capacity by 4, to obtain one-quarter of the cubic capacity of the cylinder = 40.5 cubic cm.**

**Multiply the one-quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the sphere = 121.5 cubic cm,**

**Divide the cylinder's surface are by 4, to obtain one-quarter of the surface area of the cylinder = 40.5 square cm.**

**Multiply the one-quarter surface area by 3 to obtain the three quarter surface area of the sphere = 121.5 square cm.**

**THE AREAS AND VOLUMES**

**OF**

**OVALS AND OVOIDS**

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

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.**

**Given that the diameter of the ball in the diagram above is 6 cm, the area of the square of the diameter will be 36 square cm; the three-quarter square area of the circle will be 27 square meters.**

**If the diameter length is then compressed down to 3 cm:**

**The area of the oval will still be 27 square cm; and be three-quarters of the area of the oval's rectangle.**

**Therefore we can say**

**As a circle is three-quarters of its square, its sphere is three-quarters of its cylinder**

**As an oval is three-quarters of its rectangle; its ovoid is three-quarters of its cylinder.**

THE NUMBER OF DEGREES OF LATITUDE AND LONGITUDE

THE NUMBER OF DEGREES OF LATITUDE AND LONGITUDE

**TO THE SURFACE OF A SPHERE**

A protractor has 180 Degrees to the "Height of its Diameter."

A protractor has 180 Degrees to the "Height of its Diameter."

**Therefore, a sphere has 180 Lateral Circles of 360 degrees to the "Height of its Diameter."**

**Therefore, given one full rotation of 360 degrees, a sphere equates to having**

**80 Vertical x 360 Degrees of Lateral = 64, 800 square degrees to its surface area.**

THE AREA OF 1 DEGREE

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

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

**⁰**

**of height to a protractor by 360**

**⁰**

**.**

**This served to demonstrate, that there are 180 lateral circles of 360**

**⁰**

**to the height of a sphere; therefore, defining are 64, 800**

**⁰**

**to the 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.

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.**

**Prove 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 IRRATIONALITY

OfTHE IRRATIONALITY

Of

**π**

Oxford English Dictionary (OED)

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**

TWO AREAS: ONE BLACK ONE ORANGE THREE AREAS: ONE BLACK ONE RED ONE ORANGE

TWO AREAS: ONE BLACK ONE ORANGE THREE AREAS: 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, 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 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.**

****

****

*SYMMETRICAL RATIONALITY*

**Card A Card B**

**Card B Card A**

Card B Card A Card A Card B

Card B Card A Card A Card B

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

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

**Card A. has 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.**

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.**

**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.**

Reference

Reference

**Ancient Babylonians Used Geometry To Track Jupiter Thousands Of Years Ago**

**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.

But while the Babylonians were very meticulous astronomers, there are no texts in which they appear to try to understand the peculiar motion of the planets in the sky, suggesting they didn't know (or perhaps care) about what they were looking at with regards to Jupiter.

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.

But while the Babylonians were very meticulous astronomers, there are no texts in which they appear to try to understand the peculiar motion of the planets in the sky, suggesting they didn't know (or perhaps care) about what they were looking at with regards to Jupiter.

**“They did not talk about why this happens,” said Dr. Ossendrijver. “We [now] know that this is because planets circle the Sun, and as the Earth overtakes Jupiter it creates this motion in the sky. It’s a projection effect.”**

Note

Quote:

Note

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**But while the Babylonians were very meticulous astronomers, there are no texts in which they appear to try to understand the peculiar motion of the planets in the sky, suggesting they didn't know (or perhaps care) about what they were looking at with regards to Jupiter.**

Rhetorical question: Why is it that academics feel the need to make such unqualified such unqualified statements!

What should have been said is that no tablets have been discovered as yet that show that they understood the motion of the planets in the sky or that the planets orbited the sun.

And in this regard, if these academics applied a little logic and reasoning to the matter, and given the fact that it was the Sumerians who first recognized that the Earth rotated around its own axis relative to the fixed position of the sun.

And indeed it was they who were the inventors of differential geometry, which today's mathematicians and geometer's still fail to understand 3000 years later.

It is highly unlikely that given their "far greater amount of knowledge" of mathematics - geometry - differential geometry and astronomy, that they did not understand that the planets including the Earth orbit the sun.

Rhetorical question: Why is it that academics feel the need to make such unqualified such unqualified statements!

What should have been said is that no tablets have been discovered as yet that show that they understood the motion of the planets in the sky or that the planets orbited the sun.

And in this regard, if these academics applied a little logic and reasoning to the matter, and given the fact that it was the Sumerians who first recognized that the Earth rotated around its own axis relative to the fixed position of the sun.

And indeed it was they who were the inventors of differential geometry, which today's mathematicians and geometer's still fail to understand 3000 years later.

It is highly unlikely that given their "far greater amount of knowledge" of mathematics - geometry - differential geometry and astronomy, that they did not understand that the planets including the Earth orbit the sun.

**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)