RETURN TO BABYLONIAN SCIMATH


BABYLONIAN LOGARITHMS

We've seen that the ancient Babylonians did MULTIPLICATION by using THE RECIPROCAL of A MULIPLIER. For this and other purposes, they built TABLES OF RECIPROCALS.

The Technique is a procedure for finding the reciprocals of regular numbers. A regular number is one which can be expressed as a multiple of 2, 3, and 5 only, and so has a finite sexagesimal reciprocal.  Old Babylonian scribes went to a lot of trouble in school texts to make sure that students only had to find reciprocals of regular numbers.  Indeed, most ofthe time, students were just expected to find reciprocals that were in the standard table (below, giving reciprocals of regular numbers between 2 and 1,21 = 81).

For finding the reciprocals of regular numbers not in the table, the students used a standard procedure, first explained by A.Sachs and called The Technique.  Here we briefly describe the algorithm and give some examples.

For the convenience of the modern reader, we give an utterly a historical justification of the Technique in modern algebraic terms. The basic idea is to write the reciprocal of r as a product of two terms, one of which we already know because it is in the table.  In order to do this, we first write r as a sum r = x + y, where x is a number from the standard reciprocal table.  Then we notice that

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Because x was in the standard reciprocal table, we know 1/x, and we are left to find the reciprocal of (1+y(1/x)), which is hopefully simpler.  If it is in the table, we are done, otherwise we repeat the process.

Let us go through a simple example, using the favorite Old Babylonian number 2,5 (= 125).   This number is called the igum.  Its reciprocal, which we want to find, is called the igibum.  According to the technique, we want to write 2, 5 as the sum of two numbers, one of them from the standard table.  The Babylonians 'broke off' the largest number that was in the table, in this case 5.  The reciprocal of 5 is 12. Multiply 12 into (the remaining) 2 to get 24. Add 1, you will see 25.  The reciprocal of 25 is 2,24.  Multiply 2,24 by 12.  You will see 28,48.  The  igibum is 28,48.

As a step-by-step procedure, we proceed as follows:
Step 0: Given a regular number. (2,5)
Step 1: Break off the largest number in the standard reciprocal table.(5)
Step 2: Find its reciprocal. (12)
Step 3: Multiply this number by the remainder of the original number.(12 times 2 is 24)
Step 4: Add 1.  (1 plus 24 is 25 )
Step 5: Find the reciprocal of this number (repeat steps 1 to 4 if necessary) (reciprocal of 25 is 2,24)
Step 6: Multiply the original reciprocal by this one. (2,24times 12 is 28,48)

The Technique is well-described in the tablet VAT 6505, published by Neugebauer in MKT 1, 270ff.  The tablet is somewhat broken and not all the problems can be restored.

The standard Old Babylonian reciprocal table gave the reciprocals of regular sexagesimal numbers between 2 and 81. (A number is called regular if its reciprocal is a terminating sexagesimal number.) Of the 50 or so known texts, most are incomplete and many are damaged. Some of the tablets also have a couple of extra lines at the beginning.

The table below gives the complete standard list and is taken from A. Aaboe, Episodes from the Early History of Mathematics, Fig 1.2.

The standard format for each line is "igi-n-gál-bi 1/n",where n and 1/n are the reciprocal pair of numbers.

	2	30		16	3, 45		44	1, 21

	3	20		18	3, 20		45	1, 20

	4	15		20	3		48	1, 15

	5	12		24	2, 30		50	1, 12

	6	10		25	2, 24		54	1, 6, 40

	8	7, 30		27	2, 13, 20	56	1, 15

	9	6, 40		30	2

       10	6		32	1, 52, 30

       12	5		36	1, 40

       15	4		40	1, 30

On another tablet:

	2	1

	4	2

	8	3

       16	4

       32       5

       64       6

In this last table, we easily recognize an arithmetic progression in one column (the left one) and a geometric progression in the other (the right one). Cued, we recognize an approxmation to these progressions in the first table.

The Babylonians knew about these progressions long before they were used by Pythagoras (c. 580-496 B.C.) to construct the chromatic scale of Western music. (Scholars believe that Pythagoras learned these progressions -- along with "The Pythagorean Formula" -- during his travels in Babylonia.)

Kline, in his book (p. 10) says, "They [Babylonian scribes] summed arithmetic and geometric progressions".

However, more than this is implicit here. THE COORDINATION BETWEEN A GEOMETRIC PROGRESSION AND AN ARITHMETIC PROGRESSION IS THE BASIS OF LOGARITHMS! THE ANCIENT BABYLONIANS WERE USING LOGARITHMS FOR ASTRONOMICAL CALCULATIONS!

(The great French mathematician and mathematical astronomer, Pierre Simon de Laplace (1749-1827), said that THE INVENTION OF LOGARITHMS "doubled the life of astronomers", simplifying some horrendous calculations -- turning MULTIPLICATION INTO DIVISION, DIVISION INTO SUBTRACTION, EXPONENTIATION INTO MULTIPLICATION, EXTRACTION OF ROOTS INTO DIVISION.)

The consensus of scholars is that the Babylonians had the essentials of LOGARITHMS. But these scholars seem not to have thought of another possibility:

I'm suggesting that those TABLETS be searched for possible evidence of calculation of HARMONIC PROGRESSIONS -- and, if found, evidence of their use!

Honor Assignment: "They knew that certain equation solutions reduced to log tables based on a non repeating fraction that they approximated as 2.43 in base 60 (163/60 or 2.716666.. in base 10). This is the base to the natural logarithm "e"." Now, I know that