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<<TOC>>

***Preface***
[gold] Here is some TCL scripts on Babylonian Babylonian quarter square multiplication algorithm. In reference to modern mathematics, the Babylonian  multiplication algorithms and modern extensions fall under the broad umbrella of the binomial theorem and algebra . Aside from the TCL calculator shell or graphics user interface gui, separate console programs were developed to dump the quarter square tables and check the algorithms as independent TCL procedures.
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***Introduction***
The Babylonian quarter square multiplication algorithm from clay tablets was loaded into TCL scripts. The Babylonians did not use algebra notation, so the reader will have to bear some anachronisms in the pseudocode and TCL scripts. The Babylonian quarter square multiplication algorithm is defined as the method used by  Babylonian Scribes to multiply two or more numbers from a table of quarter squares, half squares, or whole squares, as a broad definition.  The bare numbers in the Babylonian  multiplication tables do not tell the complete method, as there are mental and undocumented  components of the Babylonian multiplication methods, meaning actions off the clay tablet. For that matter, some modern quarter square tables and publications did not include any instructions, complete instructions, or any explanation on how the tables were generated. Zip instructions on algorithms and code are called job security in some quarters. In modern terms, the integer number pairs in Babylonian base_60 notation that appeared on the tablet may be N,int<N*N*.25>,<N+1>,int<(N+1)*(N+1)*.25>, <N+2>,int<(N+2)*(N+2)*.25>, until a desired goal is reached. In  Babylon, the desired end goal was usually  60 for base_60, but the algorithm steps, notation, and desired goal can be transformed into base 10, base 16, or other bases for modern schemes and computer algorithms.  In reference to modern mathematics, the Babylonian  multiplication algorithms and modern extensions fall under the broad umbrella of the binomial theorem and algebra .  The 2007 paper of David McFarland provided an excellent history of the quarter square multiplication algorithm in both the ancient and modern eras. The college institutes, textbooks,  and computer companies in the USA that previously used the Babylonian mathematical ideas without credit may be glad to know that the  Babylonian copyrights and patents expired about 4000 years ago.

In the cuneiform math problems and coefficient lists on clay tablets, there are coefficient numbers which were used in determining the amount of materials and the daily work rates of the workers. In most cases, the math problem is how the coefficient was used in estimating materials, work rates, and math problems. One difficulty is determining the effective magnitude or power of the number coefficient in the base 60 notation. In cuneiform, numbers in base 60 are written on clay tablets using a notation of relative place value. For example, 20 could represent either 20*3600,20*60,20,20/60, 20/3600, or even 1/20. The basic dimensions and final tallies were presented in the cuneiform accounts on clay tablets, but some calculations, some units, and some problem answers (aw shucks!) were left off the tablet. Successive or iterated math solutions are called algorithms and the multiplication methods are some of the earliest algorithms documented, ref the bulk of math tablets were circa 1600 BCE. The TCL procedures are descendants of this idea. The Babylonians did not use algebra notation, decimal notation, or modern units, so the reader will have to bear some anachronisms in the TCL code. At least one approach for the modern reader and using modern terminology is to develop the implied algebraic equations and decimal equivalents from the cuneiform numbers. Then the TCL calculator can be run over a number of testcases to validate the algebraic equations. For the examples of presenting algorithms to modern readers, the calculator shell will give product of long back side and short front side in meters for product of field area in meters squared. The units can be changed in the internal calculator formulas as TCL code. Any convenient and consistent output units might be used like millimeters, inches, cubits, feet, or dollars to donuts. 
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*** Formulas for Quarter Squares and  Half Squares ***
From the tablets and including modern derivations and formula extensions, there are a number of formulas that can be presented for the Babylonian quarter square multiplication algorithm into TCL code. The quarter square multiplication formula for the tables is x*y = <.25* (x+y)**2> * <.25* (x-y)**2>.  Also rearranging terms, x*y = .25*{ <(x+y)**2> * < (x-y)**2>}. The half square multiplication formula for the tables may be equivalent to x*y = 0.5* <.5* (x+y)**2> * <.5* (x-y)**2>. But there is some uncertainty specific to how the Babylonian half square tables were used,  as the tablet CBS-1535 includes integers and half fractions < N, (N+.5), (N+1), (N+1+.5) ...> in base_60 notation.  The modern extension to the multiplication algorithm from the binomial theorem is a*b = 0.5*{a+b)**2 -a**2-b**2}.

The conventional Western formula for quarter square multiplication tables is a * b= (1/4)*(((a+b)**2) - ( (a-b)**2 )  ). Another  formula with variable changes seen  in modern publication variants is a * b = ((a + b)/2)**2 - ((a – b)/2)**2. In opinion for the TCL procedures, the division operation of terms by 2  and  4 should be avoided and replaced by multiplication by .5 and .25 respectively,  because of possible computer division by zero. Also for the TCL procedures, and although some original ancient formulas and modern extensions used positive integers only, the TCL equivalent formulas are written in the code as floating point typed variables , where possible. The modern reference tables used positive numbers including the extant  Babylonian tablets, so the validity of negative terms and mixed positive/negative (+,-) numbers in products would have to be checked in possible methods using the tables etc. Not a mathematical proof, but the multiplication algorithm in the initial TCL calculator shell seems working correctly with negative and mixed (+/-) numbers. 
For comparison of the Babylonian algorithm(s) with conventional Western methods, TCL code may include redundant procs, redundant calculation paths, and printout check formulas to compute product, area, check diagnostics, and relative errors. The Babylonian quarter square multiplication algorithm is believed to be derived from the Sumerian Scribal land survey practices of averaging  two opposite sides of a field to calculate the diagonals or area of a field. These averages of opposite sides were used in the approximate quadrilateral formula expr <((a+c)/2) *((b+d)/2) >, where the four contiguous sides of the field are a,b,c,&d. At least the Sumerians were averaging field sides (a+c)/2 circa 3000 BCE.  As seen in the quarter square and half square formulas, sum and difference methods in general were an interesting stable of land surveyor's riddles, ref Friberg. But the extant Babylonian algorithms and tables discussed here are probably from a  later period  circa 1600 BCE.  

*** Testcase with Quarter Square Tables ***
For trials of the quarter square table, try multiplying 7 by 4. Some formulas can be pasted into the TCL console. The sum is expr {7+4},  11. The difference is expr {7-4},  3. From lookup 1 on the  quarter square table, the quarter square or counterpart of 11 is 60. Likewise from lookup 2, the quarter square or counterpart of 3 is 2. The product from quarter square method is expr {30-2} or 28.  The check answer from the TCL console is expr { 7*4}, 28. The overhead work is 2 lookups and 3 operations, which are abbreviated <ops,1+,2-,2L,3->. 

*** Testcases with Half Square Tables ***
For the trial method for the half square table, a testcase of 7*4 was entered and a second testcase of 7.5*4.5 was entered. For testcase 1 of the half square table, try multiplying 7 by 4. The sum is expr {7+4},  11. The difference is expr {7-4},  3. From lookup 1 on the  half square table, the half square or counterpart of sum 11 is 60. From lookup 2 on the  half square table, the half square or counterpart of difference 3 is 4. The trial product  from the half square method is expr {(60.-4.)*.5} or 28. even.  The check answer from the TCL console is expr { 7*4 },  28. The overhead work is 2 lookups and 4 operations, which are abbreviated <ops,1-,2+,2L,3-,4*>. For the second trial or testcase of the half square table, try multiplying 7.5 by 4.5. The sum is expr {7.5+4.5},  12. The difference is expr {7.5-4.5},  3. From lookup 1 on the  half square table, the half square or counterpart of sum 12 is 72. From lookup 2 on the  half square table, the half square or counterpart of difference 3 is 4. The trial product  from the half square method is expr {(72.-4.)*.5} or 34?  The check answer from the TCL console is expr { 7.5*4.5 },  33.75. The overhead work is 2 lookups and 4 operations, which are abbreviated <ops,1-,2+,2L,3-,4*>. Commenting on the fourth possible operation, the Babylonians tended to use the reciprocal 1/N or multiply ½ for division, as recommended above for the TCL procedures.
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***Many Place Reciprocals have Similarity to Modern Logarithm steps ***
 
The Babylonian Market Rate Algorithm is taking 1) base_60 reciprocals of several numbers, 2) summing the reciprocals, and 3) using the sum as a multiplication factor. The Market Rate Algorithm has initial steps and similarity to multiplication of several terms using modern logarithms. For example, take the base_60 logarithms of several numbers {  50 40 30 }, resulting in the set {  6/5 6/4  2  }, summing the base_60 reciprocals  for a factor expr { 6./5+6./4+2.  }, which equals 4.7 decimal. The return reciprocal or anti-reciprocal in base_60 is expr { 60/4.7 } is 12.76 decimal and with a factor approximating average  is expr { 3.*60/4.7 } is 38.29787. The average of the original three numbers is expr {  (50+40+30.)/3. } or decimal 40.0  For a modern example using natural logarithms, take the problem 50*40*30 = 60000. Take the sum of natural logs as expr { log(50)+log(40)+log(30) } = 11.0020998 decimal. The answer is the antilog expr { 2.718281828**11.0020998 } = 60000. Note: The reciprocal  formulas in base_60 are N*R=60, N=60/R, and R=60/N.
In the Late Babylonian period, there were numerous tablets and fragments showing many place reciprocals and factoring other numbers in base_60. The many place reciprocals are reciprocals with 5 to 10 places in base_60. The most unusual tables were many place reciprocals between 1 and 2, ref Friberg and Al-Rawi. Although less common or less extant, there are tables of many place reciprocals between 2 and 4. The square root of sixty is decimal 7.74, so continuing the tables beyond integer 8 would be redundant and not be useful. The tablets with many place reciprocals between 1 and 2 are generated for numbers between 60 and 30. The tablets with many place reciprocals between 2 and 4 are generated for numbers between 30 and 15. Because the Babylonians used relative place notation like a modern slide rule, remember that the calculated many place reciprocals between 60 and 15 could been used in relative base_60 places like 20,20*60,20*3600,20*216000 , and so on. In addition, there are rare LB. fragments or tables that have numbers in reciprocal pairs in base_60 with the added glyph ibsi, which means either square or square root, ref Friberg and Al-Rawi.  In a general sense on other tablets, the term ibsi is associated with square roots, raising a number in exponent, or the reverse, but the term ibsi linked with reciprocals is curious. The minmal conclusion is that the Late Babylonian mathematicians considered the regular base_60 reciprocals involved with squares, square roots, or powers. A complete explanation of the many place reciprocal tables is not available, but it has been proven that the many place reciprocals and square tables were used in school copying exercises,  school multiplication assignments, and factoring algorithms. There are two multiplication formulas from the Old Babylonian tablets that used differences, sums, and squares. With a squint eye of the amateur, the Old Babylonian tables with many place reciprocals in base_60 look somewhat like the many place numbers (meaning digits) in the modern tables for decimal and natural logarithms. 
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***Notes on Multiplication Methods and Modern Logarithms*** 
The natural log N1 approximates ( N1-1) / N1 or kay*(( N1-1) / N1), as the first term in series. The natural log of 2 is expr log(2) = 0.6931 decimal.  The approximate natural log of 2 equals expr (2.-1.)/2.= 0.5 decimal.  Returning to the original  problem of 50*40*30 using modern base_60 logarithms, the sum of logs is expr { 0.95546+0.90096+0.830706} , 2.687126. The antilog in base_60 is expr 60**2.687126= 59995.193. The approximate natural logs are expr (50-1.)/50.=0.98, expr (40-1.)/40.=.975, expr (30-1.)/30.=0.9666. expr .98+.975+.9666=2.9216. Kind of hokey math, but the base_60 logarithms are not that far off scale from the natural logarithms, about 8 percent off. expr 2.92-2.92*.08=2.6864, expr 60**2.6864= 59817.122. Another peg point is that the  approximate natural log of 60 is expr {(60.-1)/60.} = .9833, which is the same as expr {1.-1./60}=0.9833 Note: expr log(5) 1.6094379124341003 approximated by expr [ int [expr 5**.3]] *log(5) 1.6094379124341003

The challenge would be to find out if the Babylonian mathematicians used many place reciprocals, many place squares, or factors with the two known Babylonian multiplication formulas. It would not be surprising if some odd notations or glyphs are associated  with possible use of the multiplication formulas, especially with the  Late Babylonian Astronomers. For example, there was a unique “three wedge”  glyph (<<<)  for multiplication in some Late Babylonian multiplication tables. The “three wedge”  glyph for multiplication may be a sort of brand name or associated quality signature for the modern student or readers. Maybe coincidence, but the  Late Babylonian symbol for the “Astronomy zero” was either space, one wedge (<), two wedges (<<) or three wedges (<<<) . The space between base_60 places was probably separator intent and a single wedge was probably also separator intent. The single wedge between base_60 places looks like Babylonian ten glyph, ref Greek jot. Also, a rare variant glyph for nine (9 decimal) was the “three wedge”  glyph (<<<). While the “Astronomy zero” is very rare on the Late Babylonian clay tablets, the understanding here is that “Astronomy zero” is indicative of extreme care in error free calculations by the Late Babylonian Astronomers, which would track with the advanced and remarkable many place reciprocals or squares. Another reference is tablet W23281, which has many place reciprocals at regular intervals, N1, N1+1, N1+2.  The tablet W23281 does not have logarithms, but like the modern logarithms evaluated at regular intervals,  regular intervals are needed for useful logarithm work. However, there are few cases where the Babylonian mathematicians interpolated between two base exponents for an answer, but detailed Babylonian knowledge of logarithms has not been assessed by the professionals.  In opinion, the many place reciprocals or factors were associated with the “Astronomy Group” of the Late Babylonian mathematicians. But almost all professional and cuneiform linguists have cited a lack of evidence on the association of  many place reciprocals or factors with the “Astronomy Group”. 
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** Rules of Thumb from Babylonian Tablets**
Here are some rules of thumb from Babylonian brick constants and Babylonian math problems. A sun dried brick has water content and also as hidden water. Some sun dried bricks have straw content. Straw content should not be a factor in fired bricks. Sun dried brick weight over wet and heavy brick weight equals   40/60 or second try equals 48/60. Sun dried brick weight over baked brick weight equals  50/60. Fraction 1/6 is mortar for clay wall.  Weights of wet and sun dried bricks  in table BM36776  have factors 3/2 and 6/5, per Friberg and Robson. Second and third section of factor equals   3/2. Fourth section has factor equals 6/5. Water approximates 12/60 weight of wet brick. Weight loss (water) is 10/60 of dried brick to baked brick. { 1 kus ba.an.gi4 } means 1 cubit returned or 1 cubit back, used in brickwork. The daily cartage quota is 37_30 talents per us per day equals expr {(37+30./60.)*30.} or 1125  kg per  60*6 meters   per day.  The daily cartage quota is 1_15 talent per danna per day equals expr {(1.+15./60.)*30} or 37.5 kg per 10.8 kilometers per day. 
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***Push Button Operation***
For the push buttons, the recommended procedure is push a testcase and fill frame, change first three entries etc, push solve, and then push report. Report allows copy and paste from console. For testcases in a computer session, the TCL calculator increments a new testcase number internally, eg. TC(1), TC(2)., TC(3), TC(4) etc. The testcase number is internal to the calculator and will not be printed until the report button is pushed for the current result numbers. The current result numbers will be cleared on the next solve button. For comparison of the Babylonian algorithm with conventional Western methods, TCL code will include redundant procs, calculation paths, and printout check formulas to compute product and area. 
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***Conclusions*** 

By inspection of the quarter squares  table, one can see that many of the calculated quarter squares in the table are in fact, the square of some other number. For example in the table, 10 has the quarter square QS(10)  or counterpart of 25. Yet QS(10) or 25 is the square of 5. The number N in the first column of the quarter squares table can be a prime number and integer, but generally the values of quarter squares QS(N) are not primes, in the short table version. But one can find some sneaky primes just under the surface of the quarter squares table in primes that are  QS(N)-1 or even N2**2-1. If QS(a) and QS(b) do not solve the problem of easy hand multiplication of a*b, then the set of N and QS(N) over the numberline do bring a harvest of very interesting numbers.

The  Old Babylonian multiplication formula requires three math operations as opposed to one math operation in modern logarithms, per two numbers. Considering hand calculations and omitting the table lookups, the Old Babylonian multiplication formula is about three times the work of the modern logarithm method. In using the Babylonian multiplication formula in taking base_60 squares, using many place numbers in a hand multiplication array, there must be breakpoints where the effort of the multiplication formula is less than taking the many place square by hand calculation. Given the amount of Babylonian effort that developed the  many place reciprocals in base_60 on the clay tablets, is it possible that the known  Babylonian multiplication formulas, base_60 reciprocals, and terms could be manipulated into an algorithm, formula, or even modern logarithm of fewer operations? 

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*** Table 1 ***
%|Table 1: Babylonian Quarter Squares Tables |printed in| tcl  format|% 
%|N | int <N*N*.25> | comments if any|% 
 &| 0  |   0.00000 | modern notation includes zero's|& 
 &| 1  |   0.00000 | some fractions clipped by int |& 
 &| 2  |   1.0000 | modern decimal notation |& 
 &| 3  |   2.0000 | |& 
 &| 4  |   4.0000 | |& 
 &| 5  |   6.0000 | |& 
 &| 6  |   9.0000 | |& 
 &| 7  |   12.000 | |& 
 &| 8  |   16.000 | |& 
 &| 9  |   20.000 | |& 
 &| 10  |   25.000 | |& 
 &| 11  |   30.000 | |& 
 &| 12  |   36.000 | |& 
 &| 13  |   42.000 | |& 
 &| 14  |   49.000 | |& 
 &| 15  |   56.000 | |& 
 &| 16  |   64.000 | |& 
 &| 17  |   72.000 | |& 
 &| 18  |   81.000 | |& 
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*** Table 2 ***
%|Table 2: Babylonian Half Squares Tables |printed in| tcl  format|% 
%|N | int <N*N*.5> | comments if any|% 
 &| 0  |   0.00000 |  modern notation includes zero's|& 
 &| 0.5 | 0.00000  | some fractions clipped by int  |&  
 &| 1  |   0.00000 | modern decimal notation|& 
 &| 1.5 | 1.0000  | |&  
 &| 2  |   2.0000 | |& 
 &| 2.5 | 3.0000  | |&  
 &| 3  |   4.0000 | |& 
 &| 3.5 | 6.0000  | |&  
 &| 4  |   8.0000 | |& 
 &| 4.5 | 10.000  | |&  
 &| 5  |   12.000 | |& 
 &| 5.5 | 15.000  | |&  
 &| 6  |   18.000 | |& 
 &| 6.5 | 21.000  | |&  
 &| 7  |   24.000 | |& 
 &| 7.5 | 28.000  | |&  
 &| 8  |   32.000 | |& 
 &| 8.5 | 36.000  | |&  
 &| 9  |   40.000 | |& 
 &| 9.5 | 45.000  | |&  
 &| 10  |   50.000 | |& 
 &| 10.5 | 55.000  | |&  
 &| 11  |   60.000 | |& 
 &| 11.5 | 66.000  | |&  
&| 12  |   72.000 | |& 
 &| 12.5 | 78.000  | |&  
 &| 13  |   84.000 | |& 
 &| 13.5 | 91.000  | |&  
 &| 14  |   98.000 | |& 
 &| 14.5 | 105.00  | |&  
 &| 15  |   112.00 | |& 
 &| 15.5 | 120.00  | |&  
 &| 16  |   128.00 | |& 
 &| 16.5 | 136.00  | |&  
 &| 17  |   144.00 | |& 
 &| 17.5 | 153.00  | |&  
 &| 18  |   162.00 | |& 
 &| 18.5 | 171.00  | |&  
 &| 19  |   180.00 | |& 
 &| 19.5 | 190.00  | |&  
 &| 20  |   200.00 | |& 
 &| 20.5 | 210.00  | |& 
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*** Table 3 ***
%|Table 3: Binomial_Theorem Squares Tables |printed in| tcl  format|% 
%|a | a*a from subbing a*b = 0.5*(a+b)**2 -a**2-b**2) | comments if any |% 
 &| 0  |   0.00000 |modern notation includes zero's|& 
 &| 1  |   1.0000 | modern decimal notation|& 
 &| 2  |   4.0000 | modern notation includes zero's|& 
 &| 3  |   9.0000 | |& 
 &| 4  |   16.000 | |& 
 &| 5  |   25.000 | |& 
 &| 6  |   36.000 | |& 
 &| 7  |   49.000 | |& 
 &| 8  |   64.000 | |& 
 &| 9  |   81.000 | |& 
 &| 10  |   100.00 | |&

 
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***Table, Math Terms Associated with Babylonian Trapezoids ***
%|Table , Math Terms Associated with Babylonian Trapezoids  |printed in tcl format|modern equivalent, English| comment if any|% 
&| sag ki.gu :| side earth   |  trapezoid |   |&
&| sag ki.ta :| side earth its , possessive case | upper front of the trapezoid | |  |&
&| sag an.na  :|side upper  | upper front of the trapezoid  | ||& 
&|us gi.na :| length front  | true length of the trapezoid||& 
&| sag gi.na :| side front| true front of trapezoid |  |&
&| N1 kus i-ku-lu :| N1 cubit in eats it (it or man),  read left to right | feed parameter of the trapezoid | noun derived from Sumerian SOV phrase   |&
&| sag du :| side | triangle |  |&
&|N1 arakarum :| N1 multiply | transformation coefficient, usually scale up |  |&
Note. These Math Terms cover 4 or 5 cuneiform languages, and 5000 years of history. Math Terms are best understanding of  context  and math connotation here,not blessed by linguists and cuneiform dictionaries. 
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***Testcases Section***  

**** Testcase 1 **** 
%|table 1|printed in| tcl format|% 
%| quantity| value| comment, if any|% 
&| 1:|testcase_number | |& 
&| 11.0 :|long side meters |   |&
&| 10.0 :|short front side meters| |& 
&| 1.0 :|optional meters| |& 
&| 1.1000000000000001 :|answers: ratio long over short | |&
&| 110.25 :|squared term1   | |&
&| 0.25 :|squared term2  |  |&
&| 110.0 :|check product (expr a*b) square meters |  |&
&| 110.0 :|product from algorithm square meters |  |& 
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**** Testcase 2 ****
%|table 2|printed in| tcl format|% 
%| quantity| value| comment, if any|% 
&| 2:|testcase_number | |& 
&| 61.0 :|long side meters |   |&
&| 60.0 :|short front side meters| |& 
&| 1.0 :|optional meters| |& 
&| 1.0166 :|answers: ratio long over short | |&
&| 3660.25 :|squared term1   | |&
&| 0.25 :|squared term2  |  |&
&| 3660.0 :|check product (expr a*b) square meters |  |&
&| 3660.0 :|product from algorithm square meters |  |&  
---
**** Testcase 3 ****
%|table 3|printed in| tcl format|% 
%| quantity| value| comment, if any|% 
&| 3:|testcase_number | |& 
&| 3601.0 :|long side meters |   |&
&| 3600.0 :|short front side meters| |& 
&| 1.0 :|optional meters| |& 
&| 1.00027 :|answers: ratio long over short | |&
&| 12963600.25 :|squared term1   | |&
&| 0.25 :|squared term2  |  |&
&| 12963600.0 :|check product (expr a*b) square meters |  |&
&| 12963600.0 :|product from algorithm square meters |  |&

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***Screenshots Section***
****figure 1.****
[Babylonian Multiplicatiion Algorithm and example demo eTCL calculator screenshot ] 
 
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***References:***
   * Amazing-Traces-of-a-Babylonian-Origin-in-Greek-Mathematics, 
   * Joran Friberg. Primary reference
   * google <  Babylonian trapezoid Wikipedia >
   * Wikipedia search engine < quarter square multiplication algorithm >
   * Wikipedia search engine < multiplication algorithm >
   * Wikipedia search engine < Babylonian Math >
   * Wikipedia search engine < history of lograrithms >
   * Wikipedia search engine <mesopotamian unit measurement >
   * Wolfram Alpha search engine < trapezoid  >
   * Ancient Babylonian Algorithms, Donald E. Knuth, Stanford University 
   * [Category Algorithm]
   * A Remarkable Collection of Babylonian Mathematical Texts, Joran Friberg, 2007
   * Chalmers University of Technology Gothenburg, Sweden
   * Joran Friberg and andrew george,
   * Six more mathematical cuneiform texts in the Schoyen collection, MS 3895, 
   * Amazing Traces of a Babylonian Origin in Greek Mathematics,
   * Joran Friberg
   * A Remarkable Collection of Babylonian Mathematical Texts: 
   * Manuscripts in the Schøyen Collection: Cuneiform Texts I},
   * Joran Friberg
   * New-Mathematical-Cuneiform-Texts, by  Joran Friberg
   * and Farouk N.H. Al-Rawi
   * Quarter Tables Revisited: Earlier Tables, Division of Labor in Table Construction, and Later
   * Implementations in Analog Computers, primary reference.
   * 2007 paper of David McFarland
   * Yvonne Dold-Samplonius et al (eds), From China to Paris: 2000 Years
   * Transmission of Mathematical Ideas. (Boethius, 46). Stuttgart: Steiner, 2002.
   * web posting by rjlipton, Logarithms, Tables, and Babylonians, dec7,21010,very good ref.
   * Neugebauer and Sachs, discussions,  Tablet CBS-1535, in the Catalog of the Babylonian
   * Section, University of Pennsylvania Museum
   * A reconstruction of Plassmann’s table of quarter-squares (1933),
   * Denis Roegel
   * A Digital Quarter Square Multiplier, Everett Johnson, IEE Transactions , vol-29,n0 3, 3mar1980
   * Mathematics-in-Ancient-Iraq-A-Social-History, E. Robson
   * Neugebauer, Otto, and A. Sachs 1945. Mathematical
   * Cuneiform Texts. New Haven: American Oriental Society.m [25, page34]
   * Reference to Tablet CBS-1535 as "squares table"
   * Glaisher wrote important articles on quarter-squares.
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***Pseudocode & Equations Section*** 
======
        more than one formula for 1) tables and 2) calculator shell
        Babylonian multiplication  rule  a * b = ((a + b)/2)^2 - ((a - b)/2)^2 # used in calculator shell
        conventional Western formula for quarter square multiplication tables
        is a * b= (1/4)*(((a+b)**2) - ( (a-b)**2 )  ) # used in Western tables circa 1600 to 1950 CE.
        quarter square multiplication formula for the tables
        is x*y = <.25* (x+y)**2> * <.25* (x-y)**2>.  # recommended, avoids division by zero
        half square multiplication formula for the tables
        may be equivalent to x*y = 0.5* <.5* (x+y)**2> * <.5* (x-y)**2> # used in tables,
        # recommended, avoids division by zero
        modern extension to the Babylonian multiplication algorithm 
        from the binomial theorem is a*b = 0.5*{a+b)**2 -a**2-b**2} # considered for calculator shell
        a * b = expr { $a * $b } # TCL math.c call for check answer
        approximate Babylonian quadrilateral formula expr <((a+c)/2) *((b+d)/2) > (not exact!!!)
        product of 50*40*30 using modern base_60 logarithms,        # used in thinking pod 
        the sum of logs is expr { 0.95546+0.90096+0.830706} , 2.687126.
        The antilog in base_60 is expr 60**2.687126= 59995.193.
        For the tables, the int function is used to clip remainders to integers.
        the precision function by [mg} is used occasionally, but not every time.
        note: mental and undocumented  components in the Babylonian multiplication methods.
        For that matter, some modern quarter square tables and publications 
        did not include instructions or complete
        instructions. Zip instructions are called job security in some quarters.
======
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**Appendix Code**
*** Trial Multiplication  Calculator Shell     ***
======
        ######))))))  TCL source code follows
        # pretty print from autoindent and ased editor
        # Babylonian Multiplication Algorithm calculator
        # written on Windows XP on eTCL
        # working under TCL version 8.6
        # This code is copyrighted same as TCL version 8.6
        # 5sep2018 copyrighted under and same as TCL license.
        # Editorial rights are reserved under TCL license
        # gold on TCL club, 5sep2018
        package require Tk
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        frame .frame -relief flat -bg aquamarine4
        pack .frame -side top -fill y -anchor center
        set names {{} { long side meters  :} }
        lappend names { short front side meters :}
        lappend names { optional meters : }
        lappend names { answers: ratio long over short : }
        lappend names { squared term1 :}
        lappend names { squared term2 : }
        lappend names { check product (expr a*b) square meters : }
        lappend names { product from algorithm square meters :}
        foreach i {1 2 3 4 5 6 7 8} {
            label .frame.label$i -text [lindex $names $i] -anchor e
            entry .frame.entry$i -width 35 -textvariable side$i
            grid .frame.label$i .frame.entry$i -sticky ew -pady 2 -padx 1 }
        proc about {} {
            set msg "Calculator for Babylonian Multiplication Algorithm
            from TCL "
            tk_messageBox -title "About" -message $msg }
        proc calculate {     } {
            global answer2
            global side1 side2 side3 side4 side5
            global side6 side7 side8
            global testcase_number
            incr testcase_number
            set side1 [* $side1 1. ]
            set side2 [* $side2 1. ]
            set side3 [* $side3 1. ]
            set side4 [* $side4 1. ]
            set side5 [* $side5 1. ]
            set side6 [* $side6 1. ]
            set side7 [* $side7 1. ]
            set side8 [* $side8 1. ]
            set long_side $side1
            set short_front_side $side2
            set height $side3
            # babylonian multiplication  rule  a * b = ((a + b)/2)^2 - ((a - b)/2)^2
            set term1  [* [+ $long_side $short_front_side ] .5 ]
            set term1  [** $term1 2 ]
            set term2  [* [- $long_side $short_front_side ] .5 ]
            set term2  [** $term2 2 ]
            # babylonian uses table lookup here
            set algorithm_result [- $term1 $term2 ]
            set check_product_tcl [expr { $long_side * $short_front_side }]
            set ratio_long_over_short [/  $long_side $short_front_side ]
            set side4 $ratio_long_over_short
            set side5 $term1
            set side6 $term2
            set side7 $check_product_tcl
            set side8 $algorithm_result
        }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$aa"
            .frame.entry2 insert 0 "$bb"
            .frame.entry3 insert 0 "$cc"
            .frame.entry4 insert 0 "$dd"
            .frame.entry5 insert 0 "$ee"
            .frame.entry6 insert 0 "$ff"
            .frame.entry7 insert 0 "$gg"
            .frame.entry8 insert 0 "$hh"
        }
        proc clearx {} {
            foreach i {1 2 3 4 5 6 7 8 } {
                .frame.entry$i delete 0 end } }
        proc reportx {} {
            global side1 side2 side3 side4 side5
            global side6 side7 side8
            global testcase_number reference_factor flag
            console show;
            console eval {.console config -bg palegreen}
            console eval {.console config -font {fixed 20 bold}}
            console eval {wm geometry . 40x20}
            console eval {wm title . " Babylonian Multiplication Algorithm Report , screen grab and paste from console 2 to texteditor"}
            console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
            puts "%|table $testcase_number|printed in| tcl format|% "
            puts "&| quantity| value| comment, if any|& "
            puts "&| $testcase_number:|testcase_number | |& "
            puts "&| $side1 :|long side meters |   |&"
            puts "&| $side2 :|short front side meters| |& "
            puts "&| $side3 :|optional meters| |& "
            puts "&| $side4 :|answers: ratio long over short | |&"
            puts "&| $side5 :|squared term1   | |&"
            puts "&| $side6 :|squared term2  |  |&"
            puts "&| $side7 :|check product (expr a*b) square meters |  |&"
            puts "&| $side8 :|product from algorithm square meters |  |&"
        }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 11.0 10.0  1.0 1.1  110.25  0.25 110. 110.}
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 61. 60. 1. 1.01  3660.25  0.25 3660. 3660. }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 3601.0  3600.0  1.0 1.0001  12963600.0  0.25  12963600. 12963600.0 }
        ::ttk::button .clearallx -text clear -command {clearx }
        ::ttk::button .about -text about -command {about}
        ::ttk::button .cons -text report -command { reportx }
        ::ttk::button .exit -text exit -command {exit}
        pack .calculator  -in .buttons -side top -padx 10 -pady 5
        pack  .clearallx .cons .about .exit .test4 .test3 .test2   -side bottom -in .buttons
        grid .frame .buttons -sticky ns -pady {0 10}
        . configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
        wm title . "Babylonian Multiplication Algorithm Calculator"
        # end of file
        # gold on TCL club, 5sep2018
        # This posting, prose, screenshots, and TCL source code is        
        # copyrighted under the TCL/TK license terms.
        # Editorial rights and disclaimers retained
        # under the TCL/TK license terms       
        # and will be defended as necessary in court.
        # end of file        
======
----
***appendix : Script on Tables for TCL console ***

======
            # pretty print from autoindent and ased editor
            # Babylonian Quarter Squares Tables 
            # written on Windows XP on TCL
            # working under TCL version 8.6
            # gold on TCL, 10oct2018
            wm title . "Babylonian Quarter Squares Tables"
            namespace path {::tcl::mathop ::tcl::mathfunc}
            set tclprecision 17
            console show
            console eval {.console config -bg palegreen}
            console eval {.console config -font {fixed 20 bold}}
            console eval {wm geometry . 40x20}
            console eval {wm title . "Babylonian Quarter Squares Tables , cut and paste from console 2"}
            console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
            proc precisionx {precision float}  {
                #  tcl:wiki:Floating-point formatting, <AM>
                # select numbers only, not used on every number.
                set x [ expr {round( 10 ** $precision * $float) / (10.0 ** $precision)} ]
                #  rounded or clipped to nearest 5ird significant figure
                set x [ format "%#.5g" $x ]
                return $x
            }
            proc list_quarter_squares { aa bb} {
                for {set i $aa} {$i<=$bb} {incr i} {
                    puts " &| $i  |   [ precisionx 3 [ int  [* .25 $i $i]    ] ] | |& "; }
                return 1}
            proc list_half_squares { aa bb} {
                for {set i $aa} {$i<=$bb} {incr i } {
                    puts " &| $i  |   [ precisionx 3 [ int  [* .5 $i $i]    ] ] | |& ";
                    puts " &| [+ $i .5 ] | [ precisionx 3 [ int  [* .5 [+ $i .5] [+ $i .5] ]  ] ]  | |&  ";}
                return 1}
            proc list_binomial_theorem { aa bb} {
                for {set i $aa} {$i<=$bb} {incr i} {
                    set a $i
                    set b $i
                    puts " &| $i  |   [ precisionx 3 [expr { 0.5*(($a+$b)**2-$a**2-$b**2) } ] ] | |& "; }
                return 1}
            set testcase_number 1
            puts "%|Table $testcase_number: Babylonian Quarter Squares Tables |printed in| tcl  format|% "
            puts "%|N | int <N*N*.25> | comments if any|% "
            list_quarter_squares 0 18
            set testcase_number 2
            puts "%|Table $testcase_number: Babylonian Half Squares Tables |printed in| tcl  format|% "
            puts "%|N | int <N*N*.5> | comments if any|% "
            list_half_squares 0 20
            set testcase_number 3
            puts "%|Table $testcase_number: Binomial_Theorem Squares Tables |printed in| tcl  format|% "
            puts "%|a | a*a from subbing a*b = 0.5*(a+b)**2 -a**2-b**2) | comments if any |% "
            list_binomial_theorem 0 10
            # gold on TCL Club, 10oct2018
            # This posting, prose, screenshots, and TCL source code is
            # copyrighted under the TCL/TK license terms.
            # Editorial rights and disclaimers retained
            # under the TCL/TK license terms
            # and will be defended as necessary in court.
            # end of file
======
----
***appendix,    ***
======
        # Tablet
======

***appendix, Golden Rectangle Dimensions for even numbered area units (burs) ***
======
                    golden rectangle ratio for 2 bur units
                    short_side_S*long_side_L = 2 bur in golden rectangle ratio
                    (1*S)*(1.61803*S)=expr 2*1800 # 4 bur
                    1.61803*(S**2)=3600 sq nindas
                    S**2= expr { 3600./1.61803 }
                    set S  [expr { sqrt (2224.927) }]
                    S=47.169,  L = 76.321 from expr { 1.61803*47.169}
                    check expr {47.169*76.321 } = 3599.985 square nindas
                    rectangle 4:3 side ratio, Length width in classic ratio.
                    1_15 or 1+1/4  in table of coefficient of the diagonal of length and width
                    a 3:4:5 rule with constant differences.
                    average=( L+W)/2  ,   deviation= (L-W)/2
======
----
======
                    golden rectangle ratio for 4 bur units
                    short_side_S*long_side_L = 4 bur in golden rectangle ratio
                    (1*S)*(1.61803*S)=expr 4*1800 # 4 bur
                    1.61803*(S**2)=7200 sq nindas
                    S**2= expr { 7200./1.61803 }
                    set S  [expr { sqrt 4449.855 }]
                    S=66.7,  L = 107.922 from expr { 1.61803*66.7}
                    check expr {66.7*107.922 } = 7198.3974 square nindas
======
----
[gold] This page is copyrighted under the TCL/TK license terms, [http://tcl.tk/software/tcltk/license.html%|%this license]. 

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