Tibetan abacus with loose stones

Tibetan Treasurer in Gyantse Castle Museum with an abacus

A Tibetan abacus with loose stones ( Tib .: rde'u rtsis ) is an arithmetic aid for performing arithmetic tasks and, in particular, for conversions of sizes of different dimensions and weights .

It was used exclusively in the tax revenue departments of the administration of the central Tibetan government or in the treasury offices of large Tibetan monasteries until 1959.

There were various textbooks on how to do calculations with the abacus with loose stones. The oldest known textbook of this type was written by Düchungpa (Tib .: 'dus byung pa ) Ananda in the 17th century. The use of the abacus with loose stones obviously goes back to the time of the Tibetan monarchy (7th-9th centuries).

For arithmetic tasks in other areas, especially in the Tibetan calendar calculation and Tibetan astronomy , another arithmetic aid was used, the Tibetan sand abacus , also known as the Tibetan sand abacus.

The computing device

The basis of the arithmetic operations were rows of stones, which were laid down in front of the calculating person from left to right, i.e. horizontally, either on the ground, on a table or on a special abacus. The abacus was sometimes divided into fields with lines to make it easier to distinguish between the individual sizes. The rows represent positions , and none of these rows of stones comprised more than 9 units. The rows of stones were placed one above the other.

Quantities were basically measured values ​​for natural produce or money, such as grain, gold, silver or hay according to traditional units of measurement.

A volume measure was used for grain. The largest unit of measurement here was a khal, which was about 18 liters. 1 khal comprised 20 bre. 1 bre comprised 6 phul (a handful). 1 phul was again divided into 120 so-called “inner pieces” (Tib .: nang gi rdog ma ). The last-mentioned size specification had no practical significance and was only used for calculation to avoid residues.

The calculating stones used were: apricot kernels ( T. kham tshig ), porcelain shards ( T. dkar-yog ), black stones ( T. rdel nag ), beans ( T. rgya sran ), sticks of ten (T. bcu shing ) , half apricot kernels (Tib .: kham tshig phyed ) and white stones (Tib .: rdel dkar ).

Example for the representation of a number: The size of a grain quantity is given as 537694 khal, 19 bre and 5 phul. This gives the following picture on the abacus with loose stones:

Kind of stones	Magnitude	amount of 537694 khal, 19 bre and 5 phul in stones
Apricot kernels ( ⭖ )	100,000 khal	⭖⭖⭖⭖⭖
Porcelain shards ( ☗ )	10,000 khal	☗☗☗
Black stones (♦)	1000 khal	♦♦♦♦♦♦♦
Beans ( ⬮ )	100 khal	⬮⬮⬮⬮⬮⬮
Ten sticks ( ❙ )	10 khal	❙❙❙❙❙❙❙❙❙
Apricot kernels (⭖)	1 khal	⭖⭖⭖⭖
Half apricot kernels (◓)	1 ⁄ 2 khal	◓
Beans ( ⬮ )	1 bre	⬮⬮⬮⬮⬮⬮⬮⬮⬮
Black stones (♦)	1 phul	♦♦♦♦♦

Addition and subtraction

For the addition in arithmetic with the abacus of numbers the word "Go" (Tib .: is ' gro ) is used. This expresses that when adding stones from a disordered set are added to an ordered set, that is to say "go" from one set to the other. To carry out the addition, it is essential that no more than nine stones may be placed in a horizontal row. This regularly leads to the fact that the billing person first reduces the amount available in order to then add correspondingly larger amounts. The execution of the arithmetic operations is regularly accompanied by a chant in which the arithmetic master presents himself what he is currently doing.

Example: Add to the amount of . ${\displaystyle 189}$ ${\displaystyle 9\cdot 7=63}$

Kind of stones	(1.) Initial quantity ( ): are to be added. ${\displaystyle 189}$ ${\displaystyle 9\cdot 7=7\cdot 9=63}$	(2.) Reduction by are to be added. ${\displaystyle 7:70}$	(3.) Reduction by are to be added. ${\displaystyle 30:100}$	(4) are added, result: . ${\displaystyle 100}$ ${\displaystyle 252}$
Beans ( ⬮ )	⬮	⬮	⬮	⬮⬮
Ten sticks ( ❙ )	❙❙❙❙❙❙❙❙	❙❙❙❙❙❙❙❙	❙❙❙❙❙	❙❙❙❙❙
Apricot kernels ( ⭖ )	⭖⭖⭖⭖⭖⭖⭖⭖⭖	⭖⭖	⭖⭖	⭖⭖

In this calculation, the following text is sung:

“(1.) Seven times nine (or) nine times seven, so sixty-three, have to go exactly. (2.) If you give 3, 6, 7 to sixty-three, it's seventy. (3.) Saying seventy, with the giving of thirty there are now one hundred that just have to go exactly. (4.) So one hundred have gone. "

While the amount to be added is usually increased by taking away from the basic amount in addition, in subtraction (Tib .: 'then ; "deduct") the subtraction of higher amounts usually results in a subsequent addition of the excess taken away.

Example: Subtract from to the amount of . ${\displaystyle 243}$ ${\displaystyle 9\cdot 8=72}$

Kind of stones	(1.) Initial quantity ( ): must be subtracted. ${\displaystyle 243}$ ${\displaystyle 8\cdot 9=9\cdot 8=72}$	(2.) Reduction by are taken away, are to be added. ${\displaystyle 70:100}$ ${\displaystyle 30}$	(3.) are added. ${\displaystyle 30}$	(4.) must be deducted, result: ${\displaystyle 2}$ ${\displaystyle 171}$
Beans ( ⬮ )	⬮⬮	⬮	⬮	⬮
Ten sticks ( ❙ )	❙❙❙❙	❙❙❙❙	❙❙❙❙❙❙❙	❙❙❙❙❙❙❙
Apricot kernels ( ⭖ )	⭖⭖⭖	⭖⭖⭖	⭖⭖⭖	⭖

In this calculation, the following text is sung:

“(1.) Eight times nine (or) nine times eight, so 72 have to be deducted exactly. (2.) With the subtraction of 70 out of a hundred there are exactly thirty, which just have to go back exactly. (3.) So thirty have gone back. (4.) Two are to be deducted exactly. So two have left. "

Conversions

Tibetan measuring device (measure of capacity ) according to Vaiḍurya dkar-po (1685). The numbers are appropriate for divination.

Tibetan measuring device (measure of capacity) for grain with the size of one bre

One of the main tasks of the Tibetan tax authorities was the conversion of sizes, especially for certain amounts of grain, which had been measured with local measuring devices. It should be noted that the bulk of the tax revenue in Tibet consisted of delivered grain. This grain was stored decentrally in storehouses and the one. and outgoing quantities reported to the tax authorities. The authorities responsible for finances used a standard size for a khal grain, which was determined with a standard measuring device. This meter was called gtan tshigs mkhar ru . However, a large number of measuring devices of different sizes have been used in different parts of Tibet.

As a typical example, the case is assumed here that the tax authorities were notified of an arrival of 155 khal grain, which was measured with a local measuring device with the size of 8 bre. For the registration of this access it was important to answer the question of how much grain was delivered to the government according to the normal measure. For this type of conversion, which was always carried out with the calculating device Abacus with loose stones, the basic work of the Düchungba describes several methods, all of which agree that calculations are not based on numbers, but that quantities of stones are rearranged. This can be explained using a method that Düchungpa himself invented and which he also uses the word “walking” (Tib .: 'gro ) to describe.

Düchungpa explains his approach as follows: If the local measuring box only holds 8 bre instead of 20 bre, this means that the reported amount is actually only 8 bre instead of 20 bre or 20 khal. or 8 khal according to the normal measure. Therefore, if you take 20 khal away from the reported initial amount of 155 khal and place 8 khal in a new amount, you get the desired amount at the end of this process. According to Düchungpa, a transition from 10 to 4 or from 5 to 2 can be expected. For the calculation with the abacus this results in the following.

Initial situation:

Kind of stones	1. Initial quantity (155)	2. Target amount
Beans ( ⬮ )	⬮
Ten sticks ( ❙ )	❙❙❙❙❙
Apricot kernels ( ⭖ )	⭖⭖⭖⭖⭖

The conversion, shown here in abbreviated form, takes place in three steps, with the arithmetic master again accompanying these operations with a chant.

Step 1

Kind of stones	Initial amount is around 100 reduced	Target amount is around 40 increased.
Beans ( ⬮ )
Ten sticks ( ❙ )	❙❙❙❙❙	❙❙❙❙
Apricot kernels ( ⭖ )	⭖⭖⭖⭖⭖

2nd step

Kind of stones	Initial amount is around 50 reduced	Target amount is around 20 increased.
Beans ( ⬮ )
Ten sticks ( ❙ )		❙❙❙❙❙❙
Apricot kernels ( ⭖ )	⭖⭖⭖⭖⭖

3rd step:

Kind of stones	Initial amount is around 5 reduced	Target amount is around 2 increased. Result: 62 khal
Beans ( ⬮ )
Ten sticks ( ❙ )		❙❙❙❙❙❙
Apricot kernels ( ⭖ )		⭖⭖

Fractions

The above-described conversion method, like all other conversion methods described by Düchungpa, could only be carried out meaningfully if the ratio of the size of the local measuring device to the size of the standard measure ( gtan tshigs mkhar ru ) could be represented in the simplest possible fractions. In its 2nd chapter, the work of Düchungpa, which deals with fractions (Tib .: zur ), gives numerous examples that the arithmetic master had to learn by heart:

"For this purpose, of the five methods of conversion, the first thing to do for the calculation with fractions is the ABC of calculations, this key to clear understanding for the clouded mind, the following:

18 bre are equal to ⁹ ⁄ ₁₀ khal.

If 17 bre and 4 ½ phul and ¹ ⁄ ₆ phul resulted, these are equal to ⁸ ⁄ ₉ khal.

17 bre and 3 phul are equal to ⁷ ⁄ ₈ khal.

16 bre and 4 phul are equal to ⁵ ⁄ ₆ khal.

16 bre are equal to ⁴ ⁄ ₅ khal.

etc. etc. "

Advanced conversion methods

Other important conversion methods to be mentioned here are: 1. The difference calculation (tib .: ngo thog spor gcog ), 2. the linking of stone quantities with subsequent addition (tib .: sngon ma'i cha 'gros ) and 3. difference calculation by means of an additional interest rate - and interest discount rates for loans ( '' bun gyi spor-gcog ).

Difference calculation

The starting point of this conversion is the difference or the difference between the sizes (Tib .: bar-khyad ) of a local measuring device and the standard measuring device. The total amount determined with the local meter is placed twice.

With the fraction, which indicates the partial ratio of this difference to the size of the standard measure (1 khal or 20 bre , measured with the gtan-tshigs mkhar-ru ), one can calculate z. B. with the method described above, from the size of the total amount determined with the local measuring device, the difference to the total amount to be calculated. This difference amount is then subtracted from the total amount measured.

So if M is the total amount measured, x is the difference between the sizes of the measuring devices and MN is the total amount to be calculated, then MN = M - M • (x / 20) is calculated for the case that the standard size is greater than the volume of the local measuring device .

Examples:

If the local measuring device has the size of 16 bre and 4 phul , the difference to one khal of the normal dimension is 3 bre and 2 phul . A new amount (a new amount of stones) is calculated from the total amount with the conversion factor ¹ ⁄ ₆ and the result is subtracted from the total amount.

If the local measuring device has the size of 17 bre and 3 phul , the difference to one khal of the normal size is 2 bre and 3 phul . A new amount (a new amount of stones) is calculated from the total amount with the conversion factor ¹ ⁄ ₈ and the result is subtracted from the total amount.

Chain of stones

Conversion of a total amount laid down as a number of stones by successively calculating interconnected partial amounts (Tib .: sngon ma'i cha 'gros ; "Going based on previous partial amounts"), which are then added.

In terms of formulas, this can be represented as follows: If M is the starting amount, MN is the amount to be calculated, V is the ratio (fraction) of the size of the local measuring device to the standard measuring device (20 khal ), and if x, y, and z are different fractions, then it reads the calculation rule MN = M • x + (M • x) • y + (M • x • y) • z. Here V = x + x • y + x • y • z must apply.

Starting from M with M • x = M1, one calculates a partial amount M1, which is placed as a set of stones to the right of M.

From this result set a second set of stones is calculated with M1 • y = M2, which is placed to the right of M1.

From M2 you create another set of stones through M2 • z, which you place to the right of M2. The result is four sets of stones, viz

M, (M • x), (M • x • y), (M • x • y • z)

or

M, M1, M2, M3.

The conversion result is then given by

M1 + M2 + M3.

Expressed in numbers e.g. B .:

${\displaystyle M,\left(M\cdot {\frac {1}{4}}\right),\left(M\cdot {\frac {1}{4}}\ \cdot {\frac {1}{2}}\right),\left(M\cdot {\frac {1}{4}}\ \cdot {\frac {1}{2}}\cdot {\frac {3}{4}}\right)}$

The result of the conversion is obtained by combining (adding) the three calculated quantities of stones, i.e.:

${\displaystyle \left(M\cdot {\frac {1}{4}}\right)+\left(M\cdot {\frac {1}{4}}\ \cdot {\frac {1}{2}}\right)+\left(M\cdot {\frac {1}{4}}\ \cdot {\frac {1}{2}}\cdot {\frac {3}{4}}\right)}$

Example:

If the size of a local measuring device is 1 broad , 1 phul and 1/3 phul of the normal dimension, the total amount M determined with the local measuring device is calculated with the fraction

x = ¹ / ₂₀ to:

M1 = M • x = M • ¹ ⁄ ₂₀ .

The result M1 is converted using the fraction y = ¹ ⁄ ₆ :

M2 = M1 • y = M1 • ¹ ⁄ ₆ .

This result M2 in turn is converted using the fraction z = ¹ ⁄ ₃ :

M3 = M2 • ¹ ⁄ ₃ .

The result of the overall calculation results from the addition of these three partial amounts M1, M2 and M3.

This conversion method is based on breaking down fractions. In the example above, this is the size ratio of the local measuring device to the standard measuring device

${\displaystyle {\frac {11}{180}}\;=\;{\frac {22}{360}}\ ={\frac {18}{360}}\ +{\frac {3}{360}}\ +{\frac {1}{360}}\ }$.

This is again

${\displaystyle {\frac {1}{20}}\;+\;{\frac {1}{20\cdot 6}}\;+\;{\frac {1}{20\cdot 6\cdot 3}}}$

According to the example above, the initial amount is 720 khal, measured with the local measuring device.

Kind of stones	(1.) Initial quantity 720 khal of the local measuring device.	(2.) M1 = 720 • 1 ⁄ 20 . Go from 20 to 1. M1 = 36	(3.) M2 = 36 • 1 ⁄ 6 . Go from 6 to 1. M2 = 6	(4.) M3 = 6 • 1 ⁄ 3 = 2. Go from 3 to 1. M3 = 2	(5.) M1 + M2 + M3 = 44
Beans ( ⬮ )	⬮⬮⬮⬮⬮⬮⬮
Ten sticks ( ❙ )	❙❙	❙❙❙			❙❙❙❙
Apricot kernels ( ⭖ )		⭖⭖⭖⭖⭖⭖	⭖⭖⭖⭖⭖⭖	⭖⭖	⭖⭖⭖⭖

Difference calculation by means of interest surcharge and interest discount rates for loans

This conversion process uses a compound interest calculation method that is familiar to Tibetan mathematicians . 'Bun denotes both the loan amount owed plus interest and compound interest as well as the interest rate . A distinction is made here between two different methods of calculating debts, namely the dbus 'bun (central Tibetan interest rate: "If there are two, one is added as interest" = 50% interest) and the gtsang' bun (interest rate for the region gTsang : "If there are three, there is one One added as interest "= 33 ¹ ⁄ ₃  % interest).

The starting point is usually the calculation of the original loan amount owed with no interest for one year or several years. The annual interest is calculated and gradually deducted from the debt amount.

To do this, multiply the amount that makes up the current level of debt at the interest rate called dbus-'bun for one year by ¹ ⁄ ₃ ( dbus' bun gcig bcag "Reduction by the interest rate of Central Tibet for one year") and subtract the result from the Starting amount, which corresponds to a multiplication by ² ⁄ ₃ . This is repeated for two years based on the previously calculated debt amount ( dbus' bun gnyis bcag "Reduction by the interest rate in Central Tibet for two years") and this is then repeated again for a possible third year ( dbus-'bun gsum bcag "Reduction by the interest rate in Central Tibet for three years ”). With the interest rate of the province gTsang (Tib .: gtsang-'bun ) one does not calculate with the fraction ¹ ⁄ ₃ but with ¹ ⁄ ₄ .

For the conversion of the sizes of grain quantities, this method is then z. This can be used, for example, if the size of the local measuring device or the difference between this amount and one khal of the standard corresponds to a fraction that occurs in the compound interest calculation. For example, ³ ⁄ ₄ • ³ ⁄ ₄ • ³ ⁄ ₄ corresponds to size 8 bre + 2 phul + ¹ ⁄ ₂ + ¹ ⁄ ₈ phul and ² ⁄ ₃ • ² ⁄ ₃ corresponds to size 8 bre + 5 phul + ¹ ⁄ ₃ phul .

Example:

The local measuring device measures 1 khal , 8 bre , 5 phul and ¹ ⁄ ₃ phul of the normal measure (1 khal or 20 bre , measured with the gtan-tshigs mkhar-ru ). The total amount (M) determined with the local meter is placed twice. The amount placed second is reduced by a double interest discount. Calculated successively in Tibetan this means MD1 = M - M • ¹ ⁄ ₃ , MD2 = MD1 - MD1 • ¹ ⁄ ₃ . Calculated directly: MD2 = M • ² ⁄ ₃ • ² ⁄ ₃ . This gives the difference. This difference MD2 is added to the amount M placed first.

Verification: The difference between the size of the local measuring device and the standard size is 8 bre , 5 phul and ¹ ⁄ ₃ phul . The calculation for the sample is M = 1 khal . This results in 1 • ² ⁄ ₃ • ² ⁄ ₃ = ⁴ ⁄ ₉ khal = ⁸⁰ ⁄ ₉ bre = 8 bre + ⁸ ⁄ ₉ bre = 8 bre + ⁴⁸ ⁄ ₉ phul = 8 bre + 5 phul + ³ ⁄ ₉ phul = 8 bre + 5 phul + ¹ ⁄ ₃ phul .

See also

• Calculating on lines
• Salamin table

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.
• 'Dus byung pa Ananda: mKhas dbang' dus byung pa'i rde'u'i rtsis gzhung sarga brgyad la dag ther byas pa rab sbyangs gser gyi me long . Tibetan block print (17th century), 16 sheets and 4 sheets of tables. (Basic work of 'Du byung pa Ananda on arithmetic with the abacus with loose stones. The most important textbook known to date on arithmetic with this abacus, written in the 17th century.)
• Ngag dbang chos 'byor: mKhas dbang' dus 'byung pas mdzad pa'i rdel-rtsis gzhung sarga brgyad kyi zab gnas sgo brgya' byed pa'i 'grel bshad dper brjod' phrul gyi lde-mig . Tibetan block print (18th century), 48 sheets. (Commentary on Rab sbyangs gser gyi me long des' Dus byung pa Ananda on calculating with the abacus with loose stones.)