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THE LEGEND OF GREEK ERATOSTHENOS.


 But Jesus beheld them, and said unto them,
  " With men this is impossible;
 but with God all things are possible."

INTRODUCTION.

In the year 430 b.c. the Greek city of Athens was suffering under a great plaque. The Athenians appealed to the oracle at Delos to abate the pestilence, and they were then promised that the plaque would be taken away if the citizens double the size of one of the altars, without sacrificing the cubical form. At the time the only measuring instruments allowed was the compass and an unmarked straightedge. Needless to say the Athenians where deluded and the task could not be accomplished.

Nearly two and a half thousand years later, man is still deluded and the worldwide epidemic of irrationality continues. The only 'progress' is that natural man can now 'prove' that what was requested by the oracle is 'impossible'. It is this ignorance of present day 'science' that has motivated natural man to launch a myopic telescope into orbit around the earth.

Before we proceed let us first establish the real meaning of the task that was given to the citizens of Athens, the task of doubling the size of one of the altars, without sacrificing the cubical form.


THE TASK

The task was to construct a cube, double the volume of a given cube, using only a straightedge and compass. Quoting from the Wikipedia "Doubling the cube" :

""A significant development in finding a solution to the problem was the discovery by Hippocrates of Chios that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that  a : r = r : s = s : 2a " "

We have already show this to be true by means of the Pythagoras Theorem.
 Reference: ( 12/17= 17/24 = 24/34 ------------------(2)
 
All that remains is to establish the means of construction with an unmarked straightedge and compass. In order to do this we need to have clarity on the theory of numbers. 

NUMBERS AND THE NUMBER LINE.

Numbers are quantitative symbols and are therefore used to signify magnitudes. The only way we can equate numbers graphically is to use the geometric number line. The number line is simply a straight finite line that represents quantities on a one dimensional scale, graphically. The line is an abstraction that can be used as a measuring instrument.

The line represents the whole, and segments of the line represent the parts that make up the whole. The line is complete (absolute) and there are no 'holes' that cannot be accounted for. There is only one whole number: ONE, the complete line represents ONE. Parts of the line can be expressed as ratios of the whole, and we refer to them as rational numbers. The idea of 'whole numbers'(plural), also referred to as 'improper fractions' is a contradiction in terms. The quantity of numbers(segments) assigned to the number line will depend on the application.

In addition to indicating segments of a line as ratios with respect to the whole, we can also indicate any position on the line by means of two sequential numbers. So for example; 12=13 will indicate the position on the diagonal of a square, that represents the extension of one side of the square. It simply means that the length of the side of a square is shown where 12 ends and 13 begins on the diagonal. In terms of magnitude the length of the side is 12/17, a rational number, 12 being the part, and 17 representing the whole, ONE.

 SUMMARY:
There is only one whole number, One(1), all other numbers are rational numbers, and they are all part of One. A rational number has two elements, the numerator(part) and the denominator(whole). The first numerator is the small one.

  THE MATRIX

A matrix can be defined as " A rectangular arrangement of quantities  in rows and columns that is manipulated according to certain rules" (Oxford Dict.)
We know that the present numerical matrix is irrational in that it is open ended and contains irrational(improper) fractions, as well as the pseudo number "zero". We will now examine the rational number matrix(base ten) which is a closed system based on first principles.

The Rational Number  Matrix
 (Base Ten  X=Ten)
 


 
#            a   //    a-1   //   a-2  //      a-3   //    a-4

         a-1 ^2
  a-4 * a-1^2 = a^3
       ^ IMPROPER FRACTIONS ( NUM > DEN ) ^

1        12/12       12/12          12/12            12/12             12/12 Unity
Unity
2        12/17       12/17          17/24            24/34              2/3
     144/289 = 1/2(+1)
              2/3 * 1/2 = 1/3
3        12/21         4/7             7/12             12/21              3/5   
      16/49   = 1/3(+1)
             3/5 * 1/3 = 1/5
4        12/24         1/2             2/4                 4/8                4/8
         1/4     = 1/4   
             4/8 * 1/4 = 1/8
5        12/27         4/9             9/1X           1X/45              5/11 
       6/81    = 1/5(+1)
            5/11 * 1/5 = 1/11  
6        12/2X        2/5             5/12            12/2X              6/15
        4/25    = 1/6(+1)
            6/15 * 1/6 = 1/15
7        12/32        3/8             8/21             21/56               7/19
        9/64    = 1/7(+1) 
           7/19 * 1/7 = 1/19
7 1/2  12/33        4/11           11/2X           2X/81         7 1/2/21   
    16/121   = 1/7 1/2
     7 1/2/21 * 1/7 1/2 = 1/21    2
8       12/34         6/17           17/48           48/136            8/23 
    36/289   = 1/8(+1)
           8/23  * 1/8 = 1/23
9       12/36         1/3               3/9               9/27              9/27
       1/9       = 1/9   
          9/27  * 1/9 = 1/27
X      12/38         6/19           19/5X          5X/18X          X/32  
      6/361   = 1/X(+1) 
        X/32   * 1/X = 1/32
11     12/3X        3/X              X/33            33/1X           11/37 
      9/9X    = 1/11(+1)
        11/37  * 1/11= 1/37   
12     12/42         2/7               7/24            24/84            12/42 
      4/49      = 1/12(+1)
        12/42   * 1/12= 1/42       1
       Matrix breaks down from here



                                                                           

                                             

 
   Notes:     
 

      1. a // 12 which represent unity in that a cube has twelve sides.

      2. a-1 is "a" reduced to smallest number.
           3. a-2 is the second proportion.
       4. a-3 is the third proportion (progressive multiple of a-1).
       5. a-4 is derived from a and is a cumulative numerator rational number.
       6. a-4 reflects the square(numerator) as well as the cube(denominator) values.
       7. X is ten (Base Ten).
       8. The doubling of the cube is shown in red.

     9. The 'number 0(zero)' is a pseudo number and is excluded from the matrix.
      X. The symbol " // " means "can be shown as" .  
      11. The  "="  sign is used after an operation ( *, /, -,+,) as well as for a valid proportion.

     12. A valid proportion(equation) is always sequential.
      13. a-4(7) is derived from a as follows: 12/32 = 7/19 therefore x = 19 and  12*19 = 7*32, 224=228, 56=57  a valid proportion.

     14. The matrix is a closed rational system.
      15. Notice that column A-1 has 6 pairs of numerator ratios, 1-2-3-4-6 and 12.
      16. The 12/33 line has been inserted to show the duplication of the cube.
      17. THE MATRIX SHOWS WITHOUT ANY DOUBT THAT THERE IS ONLY ONE WHOLE NUMBER, ONE, AND THAT ALL OTHER
            NUMBERS ARE RATIONAL NUMBERS(PART OF ONE).

   


 
 DOUBLING THE CUBE
 
The respective ratios of the sides of  the two cubes are 12/42(4/14) and 12/33(4/11).
 Given any cube, we mark off 4 equal segments on the side of the cube and extend the side to 14 segments by means of a compass and straight edge.  Next we sub divide the 14 segment extension  into 11 segment and mark of 4 segments of the 11 segments for the larger cube. We have constructed a cube with double the volume of a given cube. The curse is removed, the citizens of Athens are free.