Tibetan sand tobacco

The astronomer Pelgön Thrinle reckons with the sand abacus.

The Tibetan regent and astronomer Sanggye Gyatsho with a sand tobacco

A Tibetan Sanda Baku ( Tibetan ས་ གཞོང Wylie sa gzhong called), and Tibetan sand abacus is a calculation tool to perform computing tasks for calculations for Tibetan calendar and Tibetan astronomy .

It was used exclusively for the Tibetan calendar calculation and Tibetan astronomy. Its use, like the astronomy of kālacakratantra known in Tibet since the 11th century , has Indian origins. Its use in Tibet is at least a thousand years old. The sand abacus is still used by Tibetan astronomers to calculate the Tibetan calendar.

The Tibetan abacus with loose stones was used for arithmetic tasks in other areas, especially in the financial management of the Tibetan government .

The computing device

Tibetan sand calculator (based on an illustration in the Vaidurya Karpo (1685) by Desi Sanggye Gyatsho )

Tibetan sand calculator (based on an illustration in the Vaidurya Karpo (1685) by Desi Sanggye Gyatsho )

The Tibetan sand abacus was a flat board, the edge of which was provided with a narrow strip so that the sand could not fall off the board. One end of the board was rounded, while the other, just cut, had a kind of pocket in which to store the sand.

Before starting the calculations, the very fine sand was evenly distributed on the board. A sharpened wooden pen called sa thur was used to write . The numbers scratched in the sand could easily be wiped away with the thumb and new numbers written in their place. This was essential for performing the arithmetic operations. Accordingly, “wiping away” (Tib .: byis pa ; dbyi ba , bsubs pa , dor ba ) is listed as a mathematical arithmetic operation in the basic arithmetic operations . The numbers were written in the usual form of Tibetan numerals shown below .

Calculation instructions and basic arithmetic operations

Astronomical calculations are basically carried out according to a sequence of arithmetic instructions that are similar to computer programs . In addition to the already mentioned deleting or wiping away numbers (tib .: grangs ), the most important instructions for this programming language are the following:

1. Placing natural numbers on certain places (tib .: gnas ) of the sand abacus, which is equivalent to writing down (tib .: 'dri ba ) numbers. The value of these places is always ≠ 10. Places of this type are generally arranged one below the other, i.e. from top to bottom.
2. Convert (Tib .: bsgril ba ) the natural number exceeding the place value in one place to the next higher place. This is always necessary if, due to a mathematical operation (e.g. a multiplication ), the number noted on the respective position is greater than the place value. This process is actually nothing more than dividing a number by the place value and then adding the result (without remainder) to the next higher position, while the remainder remains at the specified position.
3. add (tib .: bsnan pa ) (of natural numbers)
4. subtract (Tib .: 'phri ba ) (from natural numbers)
5. multiply (tib .: bsgyur ba ) (from natural numbers)
6. divide (tib .: bgo ba ) (from natural numbers)

Tibetan treatises on astronomy treat these arithmetic operations very briefly, if at all. A student of Tibetan astronomy therefore relied on a teacher to teach him how to perform these operations through oral instruction.

Placing or writing numbers

Tibetan numbers

Numbers were noted on the sandabacus with the Tibetan digits or numerals listed on the right, but neither these digits nor the corresponding Tibetan numerals appear in the arithmetic instructions of Tibetan astronomy . Rather, so-called symbolic numerals are used.

Examples (only one selection here) for symbolic numerals:

• For the number 0: emptiness (tib .: stong pa ) or heaven (tib .: nam mkha ).
• For the number 1: hare (tib .: ri bong ), moon (tib .: zla ba ), body (tib .: gzug ) or rhinoceros (tib .: bse ru ).
• For number 2: hands (tib .: lag ), eyes (tib .: mig ), couple (tib .: tongue ) or gait (of the sun) (tib .: bgrod ).
• For the number 3: world (tib .: 'jig rten ), tip (tib .: rtse mo ) or fire (tib .: me ).
• For the number 4: ocean (tib .: rgya mtsho ), river (tib .: chu bo ), devil (tib .: bdud ) or foot (tib .: rkang ).
• For the number 5: sense organ (tib .: dbang po ), element (tib .: 'byung ba ) or arrow (tib .: mda )
• For the number 6: taste (tib .: ro ), season (tib .: dus ) or living beings (tib .: 'gro ba ).
• For the number 7: preciousness (tib .: rin chen ), wise man (tib .: thub pa ) or planet (tib .: gza ).
• For the number 8: God (tib .: lha ), desire (tib .: sred pa ) or happiness (tib .: bkra shis ).
• For the number 9: root (tib .: rtsa ), treasure (tib .: gter ) or hole (tib .: bu ga ).
• For the number 10: direction (tib .: phyogs ), strength (tib .: stobs ) or wealth (tib .: 'byor ba ).

etc. etc.

Please note that the numbers shown in this way were always written from right to left. The number given in the Tibetan arithmetic rules

"Arrow (5) Taste (6)" is = 65,

the number

"Root (9) Season (6) Eye (2) Planet (7)" = 7269.

The arithmetic rule “multiply 65 by 7269” appears in the Tibetan arithmetic instructions accordingly as

multiply "arrow taste" by "root season eye planet".

A Tibetan astronomer first had to memorize numerous symbolic numerals in order to perform calculations on the sand abacus and to observe the special writing rule for numbers.

Adding multi-digit natural numbers

As a prerequisite for the addition with the abacus, the astronomer has to add one-digit numbers, such as B. 6 + 9, 2 + 3, etc., learned by heart. As an example for performing the addition with the abacus, the task 11 + 68 + 89 is assumed. To do this, write the three numbers one below the other and proceed according to the following procedure, whereby only one column appears on the abacus, which is changed by wiping away and adding numbers.

(1.) Task: 11, 68 and 89 are to be added	(2.) 9 + 8 (will be repaid) = 17 (will be noted).	(3.) 7 + 1 (will be repaid) = 8 (will be noted).	(4.) 8 + 1 (will be repaid) = 9 (will be noted).	(5.) 9 + 6 (will be repaid) = 15 (will be noted).	(6.) 5 + 1 (will be repaid) = 6 (will be noted).
11	11	1 8	1 8	1 1 8	168
68	6 7	6_	6_	5 _
89	8th_	8th_	9 _
	1 _	1 _

Subtract multi-digit natural numbers

As a prerequisite for subtracting with the abacus, the astronomer has to subtract single-digit numbers, such as B. 10 - 9, 4 - 2 etc., learned by heart. Task 1111 - 707 is assumed here as an example for performing the subtraction. Minuend and subtrahend are written one below the other.

(1.) Task: subtract 707 from 1111	(2.) 10 - 7 = 3. Repayments 1 and 7. Note 3 (to be added)	(3.) Add 3	(4.) 10 - 7 = 3. Repayments 1 and 7. Note 0 and 3 (to be added)	(5.) Add 3
1111	111	4 11	40 1	404
707	3 07	7th	3

Multiplying multi-digit natural numbers

Tibetan multiplication table for the multiplication table based on a block print from the 17th century

As a prerequisite for the implementation of multi-digit multiplication numbers astronomer has the small multiplication table to learn by heart. The corresponding multiplication table is called the “end of nine” (Tib .: dgu mtha ) in Tibetan , since such tables start with 9 • 1 and end at 2 • 10.

The multiplication table opposite begins with the performance of its Tibetan name: dgu mtha ri'u mig “Table with the end of nine”. In the first line it lists the multiplier and multiplicand, which are written on top of each other. Below, in the second row, the result is noted.

The numbers in the first two rows of the multiplication table:

1st line (above, multiplier)	9	9	9	9	9	9	9	9	9	9	8th	8th	8th
1st line (below, multiplicand)	1	2	3	4th	5	6th	7th	8th	9	10	1	2	3
2nd line (product)	9	18th	27	36	45	54	63	72	81	90	8th	16	24

When multiplying multi-digit numbers, the multiplier is written to the left and the multiplicand to the right. Zeros at the end of the multiplier are immediately written to the end of the multiplicand. The last number of the multiplier is written under the highest number of the multiplicand. When performing the multiplication, start from the left.

As an example, the number 3210 is chosen as the multiplier and the number 92 as the multiplicand. The task is therefore 92 • 3210.

(1.) Multiplier 3210 and multiplicand 92 are placed	(2.) 9 • 3 = 27 (is noted)	(3.) 9 • 2 = 18 (is noted)	(4.) 9 • 1 = 9 (is noted)	(5.) 9 is repaid. 32 and the 1 move to the right	(6.) 2 • 3 = 6 (is noted)	(7.) 2 • 2 = 4 (is noted)	(8.) 2 • 1 = 2 (is noted)	(9.) 2 is repaid. 32 and the 1 move to the right	(10.) Multiplication by 0 results in 0
32 920	32 920	32 920	32 920	32 20	32 20	32 20	32 20	32 0
1 __	1 __	1 __	1 __	1 _	1 _	1 _	1 _	1
	7 ____	78 ___	789 __	789 __	789 __	789 __	7892 _	7892 _	78920
	2 _____	21 ____	21 ____	21 ____	216 ___	2164 __	2164 __	2164 __	2164 __

Then the two numbers written on top of each other, i.e. 78920 and 216400 (!), Have to be added, which is carried out according to the described method of addition. The result is 295320.

Division of multi-digit natural numbers

The division assumes knowledge of the multiplication table and the formation of differences to the numbers from 10 to 90. Task 1111101: 707 is given as a calculation example. To do this, you write dividend and divisor below each other.

	(1.) Task: 1111101: 707	(2.) 10: 7 = 1 (placed). 10 - 7 = 3. Replace 1 and 7, add 3	(3.) 3 is added and canceled	(4.) 7 × 1 = 7. 10 - 7 = 3. Repayments 1 and 7. Note 3 (to be added)	(5.) 3 is added and canceled	(6.) Write down the divisor again	(7th) 40: 7 = 5 (placed). 40 - 35 = 5th repayment 4 and 7th note 5 (to be added)	(8.) 5 is added and canceled	(9.) 7 × 5 = 35. 40 - 35 = 5. Repay 4 and 7. Note 5 (to be added)	(10.) 5 is added and canceled	(11.) Write down the divisor again
quotient		1 _____	1 _____	1 _____	1 _____	1 _____	15 ____	15 ____	15 ____	15 ____	15 ____
dividend	1111101	111101	411101	401101	404101	404101	04101	54101	50101	50601	50601
divisor	707___	307___	7___	3___		707__	507__	7__	5__		707_

etc. etc.

Performing astronomical calculations

Representation of the ecliptic according to modern astronomy. The Tibetan worldview is different from this.

Tibetan astronomy was particularly concerned with calculating the positions - called astronomical lengths - of the moon , the sun and the planets Venus , Mercury , Mars , Jupiter and Saturn . Here, in the geocentric worldview of the Tibetans, the sun is a planet . Tibetan astronomy divided, among other things, the ecliptic , i.e. the great circle that is created by projecting the apparent orbit of the sun onto the celestial sphere over the course of a year, into 27 parts, which were counted from 0 to 26. These parts of a circle, often called lunar stations or lunar houses , are called rgyu skar "stars on which one (ie the planets) walk" in Tibetan . Mathematically, this is an angle or a radian measure . The angle unit rgyu skar was divided into 60 chu tshod . One chu tshod was divided into 60 chu srang . A chu srang consisted of 6 dbugs , which in turn were divided into 67 cha shas “parts”. It becomes clear that with this system of angular dimensions the length of a planet could be determined very precisely.

The task chosen here as an example is the calculation of the mean length of the sun (= y) at the end of the fifth synodic month (x = 5) of a year. This requires the average change in the length of the sun per synodic month. This is a = 2 (rgyu skar) 10 (chu tshod) 58 (chu srang) 1 (dbugs) 17 (cha shas) . In addition, the length of the sun at the beginning of the year is required. This is given here with b = 25 (rgyu skar) 8 (chu tshod) 10 (chu srang) 4 (dbugs) 32 (cha shas) .

In the language of classical algebra - which was unknown to the Tibetan astronomer - the task is represented by the linear equation y = a • x + b. The concrete calculation remains as a task given the complex system of place values.

In Tibetan textbooks on astronomy, the implementation of this calculation is presented as a program text for the sand abacus, whereby the numerical values ​​have been added in brackets for easier understanding:

1. Put the number of the past synodic months (= x) in five places .
2. From above, multiply successively by (a =) "eye" (2), "cardinal direction" (10), "snake god sense organ" (58), "body" (1), "moon planet" (17).
3. From above, add one after the other (b =) “Suchein” (25), “Treasure” (8), “Zero body” (10), “Vedas” (4), “Teeth” (32).
4. Up Conversion using the values ​​“mountain taste” (67), “season” (6), “heaven taste” (60), “zero intermediate direction” (60), “wheel” (27).
5. The rest, after deleting the highest point, is (= y) the mean length of the sun.

The following procedure results for performing the calculation on the sand tobacco. It should be pointed out once again that there is always only one column with numbers on the sand abacus, which can be changed by wiping away and replacing numbers:

Status	(1.) Place past months (x) in five places	(2.) From above multiply by (a =) 2, 10, 58, 1, 17 one after the other	(2a.) Result of the multiplication	(3.) From above add one after the other (b =) 25, 8, 10, 4, 32	(3a.) Result of the addition	(4.) Upward conversion through the place value 67	(4a.) Conversion using the place value 6	(4b.) Conversion using the place value 60	(4c.) Conversion using the place value 60	(4c.) Conversion using the place value 27	(5.) The remnants are the mean length of the sun (y)
27	5	5 • 2	10	10 + 25	35	35	35	35	35	(35 + 1): 27	9
60	5	5 • 10	50	50 + 8	58	58	58	58	(58 + 5): 60	3	3
60	5	5 • 58	290	290 + 10	300	300	300	(300 + 1): 60	1	1	1
6th	5	5 • 1	5	5 + 4	9	9	(9 + 1): 6	4th	4th	4th	4th
67	5	5 • 17	85	85 + 32	117	117: 67	50	50	50	50	50

Calculating with numbers in the sexagesimal system

The numerical values ​​of a size specification that do not follow the decimal system are written one below the other on the sand abacus. The number noted at the respective place is always an integer in Tibetan astronomy or when calculating on the sand abacus. The places are always placed one above the other, for example for 3 rgyu-skar , 26 chu-tshod , 5 chu-srang and 4 dbugs :

• 3
• 26th
• 5
• 4th

The place values ​​in this example (from top to bottom) are 27, 60, 60 and 6. So the sexagesimal system was essentially followed . The values ​​were not noted separately in Tibet.

In order to reproduce such numbers in a more space-saving manner in the following, the number sizes are noted in square brackets separated by commas and the place values ​​after them are given in round brackets separated by a slash. The above numerical value is given as [3.26.5.4] / (27.60,60.6). Generally speaking, these mostly five-digit numbers are referred to below as

• [ , , , , ] / ( , , , , ) ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$ ${\displaystyle a_{3}}$ ${\displaystyle a_{4}}$ ${\displaystyle a_{5}}$ ${\displaystyle st_{1}}$ ${\displaystyle st_{2}}$ ${\displaystyle st_{3}}$ ${\displaystyle st_{4}}$ ${\displaystyle st_{5}}$

written, where integers and the place values ​​are. ${\displaystyle a_{n}}$ ${\displaystyle st_{n}}$

Addition and subtraction

For the addition of two multi-digit numbers one proceeds according to the rule:

[ ] / ( ) + [ ] / ( ) = ${\displaystyle a_{1},a_{2},a_{3},a_{4},a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$ ${\displaystyle b_{1},b_{2},b_{3},b_{4},b_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

[ ] / ( ). ${\displaystyle a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3},a_{4}+b_{4},a_{5}+b_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

If individual sums are greater than the place value, conversion is carried out using the procedure described below (see multiplication and conversion to place values).

The procedure for subtraction is analogous:

[ ] / ( ) - [ ] / ( ) = ${\displaystyle a_{1},a_{2},a_{3},a_{4},a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$ ${\displaystyle b_{1},b_{2},b_{3},b_{4},b_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

[ ] / ( ). ${\displaystyle a_{1}-b_{1},a_{2}-b_{2},a_{3}-b_{3},a_{4}-b_{4},a_{5}+b_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

If the amount to be deducted (subtrahend) is greater than the amount to be reduced (minuend), the amount above the minuend is reduced by one before the subtraction and the value added to the minuend. If this occurs at the highest point, this is simply increased by the value.

Multiplication and conversion to place values

The multiplication of a multi-digit size specification by a whole number c is still relatively easy. Here is the general solution to the problem that is familiar to Tibetan mathematicians:

c • [ ] / ( ) = ${\displaystyle a_{1},a_{2},a_{3},a_{4},a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

[ ] / ( ). ${\displaystyle c\cdot a_{1},c\cdot a_{2},c\cdot a_{3},c\cdot a_{4},c\cdot a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

For the Tibetan mathematicians, the solution of the multiplication of two multi-digit sizes, i.e. the solution of the problem, turned out to be much more difficult

[ ] / ( ) • [ ] / ( ), ${\displaystyle a_{1},a_{2},a_{3},a_{4},a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$ ${\displaystyle b_{1},b_{2},b_{3},b_{4},b_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

it could happen that the place values ​​of the numbers to be multiplied were also different.

Such tasks resulted from two conversion factors. The first of these two values ​​(A) indicates the ratio of the mean solar month or zodiac day to the mean synodic month or mean lunar day and is noted as follows:

[1,2] / (-, 65)

It should be noted that this factor, written as a fraction, with

A = ${\displaystyle 1+{\frac {2}{65}}={\frac {67}{65}}}$

is to be equated.

The second of these conversion factors (B) gives the ratio of the mean lunar day to the mean natural day and is noted as follows:

1 - [0,1,1] / (-, 64,707).

Written as a fraction, this corresponds to B = . ${\displaystyle 1-{\frac {1+{\frac {1}{707}}}{64}}}$

Multiply the conversion factor A by the period of revolution of the sun in zodiac days, i.e. i. with 360, one gets with the revolution time of the sun in lunar days. The relatively simple conversion with the sand abacus to the value of the time unit lunar day then results in the following:

• 360 • [1,2] / (-, 65) =

• [360 • 1, 360 • 2] / (-, 65) =

• [360, 720] / (-, 65) =

• [371, 5] / (-, 65) =

• [371, 300] / (-, 65 • 60) =

• [371, 4, 40] / (-, 60.65) =

• [371,4, 2400] / (-, 60, 65 • 60) =

• [371, 4, 36, 60] / (-, 60, 60, 65) =

• [371, 4, 36, 360] / (-, 60,60,65 • 6) =

• [371, 4, 36, 5, 35] / (-, 60,60,6, 65) =

• [371, 4, 36, 5, 7] / (-, 60,60,6, 13).

This is the orbit of the sun on lunar days. The process of converting to place values ​​itself corresponds to expanding fractions. In the language of modern mathematics, this equates to this calculation

360 • a = 360 • 67/65 = 371.107962 lunar days.

The above calculation also provides an example of dividing a number with several digits by an integer, because it is the task

[720,0,0,0] / (60,60,6): 65 is included.

To calculate the period of revolution of the sun in natural days one now has to multiply the period of revolution of the sun in lunar days by B, which is the task for the Tibetan astronomer

[371,4,36,5,7] / (-, 60,60,6,13) • (1- [0,1,1] / (-, 64,707)) =

[371,4,36,5,7] / (-, 60,60,6,13) - [371,4,36,5,7] / (-, 60,60,6,13) • [0 , 1.1] / (-, 64.707)

revealed.

With

[371,4,36,5,7] / (-, 60,60,6,13) • [0,1,1] / (-, 64,707)

the task is to multiply two numbers with different values.

The Tibetan astronomers solved such problems by applying the problem to the problem described above

c • [ ] / ( ) = [ ] / ( ) ${\displaystyle a_{1},a_{2},a_{3},a_{4},a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$ ${\displaystyle c\cdot a_{1},c\cdot a_{2},c\cdot a_{3},c\cdot a_{4},c\cdot a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

where c is an integer.

In the present case, the amount of the revolution time of the sun of [371,4,36,5,7] / (-, 60,60,6,13) was not only applied to the smallest unit 13, but also to the tiny time unit 13 • 707 parts of the dbugs converted.

In general, this corresponds, for example, to converting a time specification from days, hours, minutes and seconds into the unit second, such as that 1 day + 0 hours + 0 minutes + 0 seconds equals 86400 seconds.

In the present case the result of the conversion is 73668268800 and the task runs on the arithmetic task

73668268800 -73668268800 • [0,1,1] / (-, 64,707) = 73668268800 - [0, 73668268800 • 1, 73668268800 • 1] / (-, 64,707) addition.

The result of this calculation is 72515574000 and is the period of revolution of the sun in natural days. ${\displaystyle {\frac {1}{60\cdot 60\cdot 6\cdot 13\cdot 707}}}$

According to the Tibetan representation on the sand tobacco, this gives the amount for the period of revolution of the sun in natural days

[0, 0, 0, 0, 0, 72515574000] / (-, 60,60,6,13,707) or after conversion

[365, 16, 14, 1, 12, 121] / (-, 60,60,6,13,707).

division

The division of a multi-digit number by an integer c is accordingly

[ ] / ( ): c = ${\displaystyle a_{1},a_{2},a_{3},a_{4},a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

[ ] / ( ) ${\displaystyle a_{1}:c,a_{2}:c,a_{3}:c,a_{4}:c,a_{5}:c}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

carried out. You start at the top. The remainder of the division of the respective higher digits are multiplied by the value of the following digit and added to the numerical value of this digit before dividing this digit.

The task turns out to be more difficult if the divisor is also a multi-digit number, as in:

[ ] / ( ): [ ] / ( ). ${\displaystyle a_{1},a_{2},a_{3},a_{4},a_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$ ${\displaystyle b_{1},b_{2},b_{3},b_{4},b_{5}}$ ${\displaystyle st_{1},st_{2},st_{3},st_{4},st_{5}}$

It could happen that the values ​​of the divisor were different from those of the dividend. Since the number of such tasks was limited, the Tibetan astronomers tried in these cases to trace the task back to a multiplication. This was done by determining the reciprocal value of a divisor C and multiplying the result by the dividend. ${\displaystyle {\frac {1}{C}}}$

Here are two examples:

1. The divisor is B = 1 - [0,1,1] / (-, 64,707).

In this case, the size was determined ${\displaystyle {\frac {1}{B}}}$

[1, 1, 1, 1] / (-, 63,696,11135).

The dividend was then multiplied with this value as a factor.

2. Let the divisor be A = [1,2] / (-, 65).

In this case the value was determined ${\displaystyle {\frac {1}{A}}}$

1 - [0, 2] / (-, 67),

by which the dividend was then to be multiplied.

literature

• Dieter Schuh: Studies on the History of Mathematics and Astronomy in Tibet, Part 1, Elementary Arithmetic . Central Asian Studies of the Department of Linguistics and Cultural Studies of Central Asia at the University of Bonn, 4, 1970, pp. 81–181
• Dieter Schuh: Studies on the history of the Tibetan calendar calculation . Wiesbaden 1973

See also

• Tibetan abacus with loose stones
• abacus
• Calculating on lines

Web links

• Encyclopaedia Cinematographica , Tibetans, Central Asia - Calendar calculation by an astronomer . (PDF) Institute for Scientific Film, Section Ethnology, Series 11, Number 32, 1981