The mathematics of the Hammurabi code

Uruk is known as the first location in history that we can call a city. An urban revolution that took the human being from the Neolithic and that became the cradle of writing, the same cradle where mathematics would also grow. It was the result of a need, not a whim. For the management of the distribution of land, salaries, control of the flooding of rivers, etc. A new lifestyle that demanded from the human being of the time a new set of knowledge and a social order that were unprecedented. Throughout several centuries this organization was forged, the Sumerians and Babylonians invented writing and perfected it. And historically falls on Hammurabi the honor to seek this unification between the lands that “reigned” with initiatives such as the well-known “Code of Hammurabi.” A set of laws in which civil and criminal aspects are confused, stripped of superstition and religious myths.

If you want to see what a society is like, go to its laws, because they do nothing more than govern daily life. By reading this code we can learn something about the mathematics of Mesopotamia, since mathematics was born precisely to improve everyday life. None of this could be imagined by those people who found a wonderful 2.4 m high basalt stele. It was in Susa (Persia) and it was the year 1901.

Shamash, the Lord of the Sun, dictated to Hammurabi the complete code for the “welfare of the people.” Although once the laws that make up the code have been wanted to be seen as soulless (law of the talion), it is far from being so. It focuses on the opposite: doing the same justice for all men. We must put on the shoes of four thousand years ago to get closer to them. It is an important step (although there are other previous compilations), as it tries to prevent citizens from taking justice into their own hands. The code has 282 laws, below a few have been chosen to account for the presence of mathematics in the lives of the inhabitants of the area at the time. References to units of measurements, quantities, proportions, interests, etc. are continuous.

1. The concept “proportionality” appears very commonly, with the operation “product” as associated:

• Law 5: If a judge has pronounced a sentence, rendered a judgment, by sealed act, and if he subsequently annuls his sentence, he will appear for such annulment, and will pay twelve times the claim that was the object of the litigation; he will be removed from office without referral and may never appear on stage.
• Law 8: If someone has stolen an ox, ram, pig, donkey, boat, from the temple or the palace, he will pay thirty times the value; if it is a nobleman, ten times the value, and if he has nothing to pay with, he will be guilty of death.
• Law 12: If the seller had died a natural death (in the interim period), the buyer will take from this seller’s house five times what he has the right to claim.
• Law 46: If the owner has not yet received his rent and has leased it for half or a third, he will share proportionally with the tenant the wheat that is in the field.
• Law 59: If someone without the permission of the owner has cut down a tree in an orchard, they will pay half a silver mine.
  [Mine is a unit of mass]

2. Also the concept of fraction:

• Law 201: If you break the teeth of a nobleman, you will also pay a third of a silver mine.
  [In other translations muskenum is used instead of noble and is quite widespread]
• Law 224: If the veterinarian of oxen and donkeys treats an ox or. an ass. of a serious illness and it cures it, the owner of the animal will give as salary to the veterinarian a sixth of silver sicle.
  [The use of “shekel” is more widespread, so from now on we substitute “sicle” for “shekel”. One mine is 60 cycles]

3. Net amounts as punishment for a wrong done:

• Law 204: If a noble hits a noble on the head, he will pay ten shekels of silver.
• Law 211: If she is a noble woman, who aborts as a result of the blows, she will pay five shekels of silver.

4. The volume measurement system (GUR) and surface area (GAN) used at the time can be seen, as well as operations between them. This can also be understood as a predecessor to the current “conversion factors” between units or even to the “currency” between currencies (from GUR to GAN, depending on the circumstances):

• Law 44: If a person has leased a vacant field for three years to cultivate it and has been negligent and has not cultivated it at all, he will return it to the owner the fourth year, after working, raking and sowing, and will give the owner ten gur wheat for ten gan.
• Law 56: If a person has released the water and flooded the nearby field, he will restore ten guru of wheat per ten gan of area.
• Law 57: If a shepherd has not come to terms with the owner of a field to graze his rams there and without an agreement with the latter he has grazed his cattle, the owner will do the harvesting himself, but the shepherd will also give him 20 gur de wheat for ten gan of area.
• Law 58: If, after the rams have left the fields and passed the gates, a shepherd has taken his herd to the field and made it graze on it, the shepherd will retain the field provided with fodder and when the harvest comes, he will It will give the owner 60 guru of wheat for 10 gan of area.

5. Appearance of temporary deadlines:

• Law 13: If this man does not have witnesses at hand, the judge will give him a period of six months, and if in this intermission he does not provide them, he acts in bad faith and deserves to be sentenced ipso facto .

6. Some laws really seem like elementary math problems, this set can be considered in any school:

• Law 60: If a person has entrusted his field to another to be converted into a garden, and the gardener has converted it into a garden and cared for it for four years, in the fifth year the owner and the gardener will split equally, but the owner will determine by his side his share.
• Law 61: If the gardener has not planted all of the trees and did not do so in the upper part of the field, this part will enter his portion.
• Law 62: If you have not turned the field entrusted to you into a garden, and the field is for cereals, you must give the owner of the field an income equivalent to the yield of the neighbor’s field, for all the years in which it had not been cultivated and you will also have to carve it and hand it over to the owner.
• Law 63: If it is an uncultivated field, he must till the field and thus hand it over to the owner, giving him ten gur of wheat per ten gan of surface each year.
• Law 64: If a person has cultivated his garden to a gardener, this, during the duration of the operation, will deliver two thirds of the rent to the owner and one third will be reserved.
• Law 228: If an architect has built a house for someone and has built it well, he will receive two shekels of silver per SAR of the house as a salary.

7. The economic administration of goods activities are appreciated:

• Law 89: If a merchant has given wheat or money at interest, he will take 100 qa per guru of wheat for it. If you have given money will perceive shekel interest in a sixth plus six is as interest.
  [qa and se are units of volume]

• Law 90: If someone has obtained credit and does not have money to repay, but does have wheat, the merchant will receive 100 qa of wheat per guru as interest.

• Law 91: If the merchant challenges and has recharged the interest of 100 qa of wheat per gur or the interest of one sixth of a shekel of silver plus six se, and has received it, he will lose all ten that he has given (borrowed).

8. The need for mathematics for the payment of wages is more than evident:

• Law 257: If someone has leased an AK-SU, they will give eight guru of wheat per year.
  [AK-SU is translated as “harvester” in other sources]
• Law 258: If someone has leased an SA-GUD, they will give six guru of wheat per year.
  [SA-GUD is translated as “cowboy” in other sources]

9. And for the liquidation of rents (we only put some):

• Law 268: If someone has leased a cargo ox, the rental price is 20 qa.
• Law 269: If you have leased a donkey as a beast of burden, the rental price is 10 qa.
• Law 270: If it is a colt or a steer, the rental price is one qa .
• Law 271: If someone has leased cart oxen with the driver, he will pay 180 qa of wheat daily.
• Law 272: If someone has leased a car for the car only, he will pay 40 qa of wheat per day.
• Law 273: If someone rents a journeyman after the beginning of the year until the fifth month, deliver on sixth day of silver; from the sixth month until the end of the year, will give a day five of silver.

Both in the prologue and in the epilogue of the code the phrase “to prevent the strong from oppressing the weak” appears. And it is that mathematics is present in the execution of laws and this has been the case between different cultures. The Greek goddess Themis, for example, represents justice. And he does it with a scale in hand. It is curious that the equations are usually taught using a balance in which the plates act as the two parts of the equality. Themis hand would be the symbol of equality. Justice and Mathematics are hand in hand.

Despite this, the last law may shock us and concur in an inconsistency with that epilogue. However, we must remember that they were written 3700 years ago. It is a society far away in time and distance, so it makes no sense to assess it with our eyes. In any case, there is the last law:

Law 282: If a slave says to his owner: “You are not my owner”, he will be made to appear as a slave, and his owner will cut off his ear.

There is an interesting variety of translations of the Hammurabi code into modern languages. In this article we have used a translation by Gabriel Franco. This text appears in the article The laws of Hammurabi: Spanish version, introduction and annotations (Journal of Social Sciences, College of Social Sciences of the University of Puerto Rico, Vol. 6.1962, 3, p. 331-356). Here you can read the 282 laws with some explanatory notes.