Le Papyrus de Rhind

Lundi 19 novembre 2012,

Traduit de l'anglais par Hervé....mon ami.

Le Papyrus de Rhind

Le Papyrus de Rhind est considéré comme la plus importante source d‘informations concernant les mathématiques des anciens Égyptiens. Ce rouleau, de 30 cm de haut sur 5,5 mètres de long, a été retrouvé dans une tombe de Thèbes sur la rive Est du Nil. Le scribe Ahmès l’a écrit en hiératique, une écriture dérivée du système hiéroglyphique. Cet écrit ayant été réalisé vers 1650 avant J.C. environ, Ahmès est l’individu le plus anciennement référencé de l'histoire des mathématiques ! Le rouleau contient également les symboles mathématiques les plus anciens connus à ce jour pour des opérations mathématiques : le signe “+” est en effet désigné par une paire de jambes allant vers le nombre à ajouter.

En 1858, l'égyptologue et avocat écossais Alexander Henry Rhind visitait l’Egypte pour soigner des problèmes de santé quand il acheta le parchemin sur un marché à Louxor. Le British Museum de Londres a acquis le rouleau en 1864.

Ahmès écrivit que le papyrus permet un "calcul précis afin d’enquêter sur les choses, et d’avoir la connaissance de toutes choses et de tous mystères ... en toute sécurité ". Le contenu du rouleau traite de problèmes mathématiques intégrant des fractions, des progressions arithmétiques, l'algèbre et la géométrie pyramidale, ainsi que des cas mathématiques utiles à l'arpentage, la construction et la comptabilité. Le problème qui m'intrigue le plus est le problème 79 dont l'interprétation a d'abord été déroutante.

Aujourd'hui, beaucoup interprètent le problème 79 comme un puzzle qui peut être traduit par «Sept maisons contiennent sept chats. Chaque chat tue sept souris. Chaque souris a mangé sept épis. Chaque épi aurait produit sept heqats (1 heqat ou hekat = 4.8 litres) de le blé. Combien cela fait-il d’heqats ?". Fait intéressant, ce mème casse-tête indestructible, impliquant le chiffre 7 et des animaux, semble avoir persisté pendant des milliers d'années ! En effet, nous observons quelque chose de similaire dans le Liber Fibonacci Abati (Livre des calculs), publié en 1202, et plus tard dans le puzzle de St. Ives, un poème pour enfants en vieil anglais qui fait référence à 7 chats.

Rhind Papyrus

The Rhind Papyrus is considered to be the most important known source of information concerning ancient Egyptians mathematics. This scroll, about a foot (30 cm) high and 18 feet (5.5 meters) long, was found in a tomb in Thebes on the east bank of the river Nile. Ahmes, the scribe, wrote it in hieratic, a script related to the hieroglyphic system. Given that the writing occurred in around 1650 B. C., this makes Ahmes the earliest-known individual in the history of mathematics! The scroll also contains the earliest-known symbols for mathematical operations - plus is denoted by a pair of legs walking toward the number to be added.
In 1858, Scottish lawyer and Egyptologist Alexander Henry Rhind had been visiting Egypt for health reasons when he bought the scroll in a market in Luxor. The British Museum in London acquired the scroll in 1864.
Ahmes wrote that the scroll gives an "accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secures." The content of the scroll concerns mathematical problems involving fractions, arithmetic progressions, algebra, and pyramid geometry, as well as practical mathematics useful for surveying, building, and accounting. The problem that intrigues me the most is Problem 79, the interpretation of which was initially baffling.
Today, many interpret Problem 79 as a puzzle, which may be translated as "Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats (measures) of wheat. What is the total number of all of these?" Interestingly, this indestructible puzzle meme, involving the number 7 and animals, seems to have persisted through thousands of years! We observe something quite similar in Fibonacci's Liber Abati (Book of Calculation), published in 1202, and later in the St. Ives puzzle, an Old English children's rhyme involving 7 cats.

Plimpton 322

Plimpton 322 refers to a mysterious Babylonian clay tablet featuring numbers in cuneiform script in a table of 4 columns and 15 rows. Eleanor Robson, a historian of science, refers to it as "one of the world's most famous mathematical artifacts." Written around 1800 B. C., the table lists Pythagoreans triples - that is, whole numbers that specify the side lengths of right triangles that are solutions to the Pythagorean theorem a^2 + b^2 = c^2. For example, the numbers 3, 4, and 5 are a Pythagorean triple. The fourth column in the table simply contains the row number. Interpretations vary as to the precise meaning of the numbers in the table, with some scholars suggesting that the numbers were solutions for students studying algebra or trigonometry-like problems.
Plimpton 322 is named after New York publisher George Plimpton who, in 1922, bought the tablet for $10 dollars from a dealer than donated the tabMesopotamia, the fertile valley of the Tigris and Euphrates rivers, which is now located in Iraq. To put the era into perspective, the unknown scribe who generated Plimpton 322lived within about a century of King Hammurabi, famous for his set of laws that included "an eye for an eye, a tooth for a tooth." According to biblical history, Abraham, who is said to have led his people west from the city of Ur on the bank of the Euphrates into Canaan, would have been another near contemporary of the scribe.
The Babylonians wrote on wet clay by pressing a stylus or wedge into the clay. In the Babylonian number system, the number 1 was written with a single stroke and the numbers 2 through 9 were written by combining multiples of a single stroke.

Magic Squares

Legends suggest that magic squares originated in China and were first mentioned in a manuscript from the time of Emperor Yu, around 2200 B.C. A magic square consists of N^2 boxes, called cells, filed with integers that are all different. The sums of the numbers in the horizontal rows, vertical columns, and main diagonal are equal.
If the integers in a magic square are the consecutive numbers from 1 to N^2, the square is said to be of the Nth order, and the magic number, or sum of each row, is a constant equal to N(N^2  + 1)/2.
As far back as 1693, the 880 different fourth-order magic squares were published posthumously in Des quassez ou tables magiques by Bernard Frenicle de Bessy, an eminent amateur french mathematician and one of the leading magic squares of all time.
We've come a long way from the simplest 3 x 3 magic squares venerated by civilizations of almost every period or continent, from the Mayans Indians o the Hasua people of Africa. Today, mathematicians study these magic objects in high dimensions - for example, in the form of fourth-dimensional hypercubes that have magic sums within all appropriate directions.

Dice

Imagine a world without random numbers. In the 1940s, the generation of statistically random number was important because was important to physicists simulating thermonuclear explosions, and today, many computer networks employ random numbers to select unbiased samples of potential voters.
Dice, originally made  from anklebones of roofed animals, were one of the earliest means for producing random numbers. In ancients civilizations, the gods were believed to control the outcome of dice tosses; thus, dice were relied upon to make crucial decisions, ranging from the selection of rulers to the division of property in an inheritance. even today, the metaphor of God controlling dice  is common, as evidence by astrophysicist Stephen Hawking's quote, "Not only God play dice, but He sometimes confuses us by throwing them where they can't be seen."
the oldest known dice were excavated together with a 5,000 year backgammon set from the legendary Burnt City in southeastern Iran. The city represents four stages of civilization that were destroyed by fires before being abandoned in 2100 B.C. At this same site, archeologists also discovered the earliest known artificial eye, which once stared out hypnotically from the face of an ancient female priestess or soothsayer.
For centuries, dice rolls have been used to teach probability. For a single roll of an n-sided die with a different number for each face, the probability of rolling any value is 1/n. The probability of ruling a particular sequence of i numbers is 1/n^i. For example, the chances of rolling a 1 followed by 4 on a traditional die is 1/6^2 = 1/36. Using two traditional dice, the probability of throwing any given sum is the number of ways to throw that sum divided by the total number of combinations, which is a sum divided by the total number of combinations, which is why a sum of 7 is much more likely than a sum of 2.

Quipu

The ancient incas used quipus (pronounced "key-poos"), memory banks made of strings and knots, for storing numbers. Until recently, the oldest-known quipus dated from about A.D.650. However, in 2005, a quipu from the Peruvian coast city of Caral was dated about 5,000 years ago. The incas of south America had a complex civilization with common state religion and a common language. Although they did not have writing, they kept extensive records encoded by logical numerical system on quipus, which varied in complexity from the three to around three thousand cords. Unfortunately, when the Spanish came to South America, they saw the strange quipus and thought they were works of the Devil. The Spanish destroyed thousands of them in the name of God, and today only 600 quipus remain. Knots types and positions, cord directions, cord levels, and color and spacing represent numbers mapped to real-world objects. Different knot groups were used for different powers of 10. The knots were probably used to record human and material resources and calendar information. The quipus may have contained more information such as construction plans, dance patterns, and even aspects of inca history. The quipu is significant because it dispels the notion that mathematics flourishes only after a civilization has developed writing; however, societies can reach advanced states without ever having developed written records. Interesting, today there are computer systems whose file managers are called quipus, in honor of this very useful ancient device. One sinister application of the quipusby the incas was as a death calculator. Yearly quotas of adults and children were ritually slaughter, and this enterprise was planned using a quipu. Some quipus represented the empire, the cords referred to roads and the knots to sacrificial victims.

Ishango bone

In 1960, Belgium geologist and explorer Jean de Heinzelin de Braucourt (1920-1998) discovered a baboon bone with markings in what is today the Democratic Republic of Congo. The Ishango bone, with it's sequences of notches, was first thought to be a simple tally stick used by a Stone Age African. However, according to some scientists, the marks suggest a mathematical prowess that goes beyond counting of objects. The bone was found in Ishango, near the headwaters of the Nile River, the home of a large population of upper Paleolithic people prior to a volcanic eruption that buried the area. One column of marks on the bone begins with three notches that double to six notches. Four notches double to eight. Ten notches halve to five. This may suggest a simple understanding of doubling and halving. Even more striking is the fact that numbers in other columns are all odd (9, 11, 13, 17, 19, 21). One column contains the prime numbers between 10 and 20, and the numbers in each column sum to 60 or 48, both multiplies 12. A number of tally sticks have been discovered that predate the Ishango bone. For example, the Switzerland Lebombo bone is a 37,000-yard baboon fibula with 29 notches. A 32,000-year old tibia with 57 notches, grouped in fives, was found in Czechoslovakia. Although quite speculative, some have hypnothized that the markings on the Ishango bone form a kind of lunar calendar for a Stone Age woman who kept their menstrual cycles, giving rise to the slogan "menstruation created mathematics." even if the Ishango was a simple bookkeeping device, these tallies seems to sets us apart from animals and represent the first steps to symbolic mathematics. The full mystery of the Ishango bone can't be solved until other similar bones are discovered.