I found this list of the contents of an ancient Egyptian maths manuscript, papyrus 10057, on the chapter on the Rhind Mathematical Papyrus dating from c.1650 BC, in Henrietta Midonick’s The Treasury of Mathematics: 1 (Harmondsworth: Penguin 1965). The book’s a collection of ancient maths texts from around the world, with relevant commentaries and explanations. I found it interesting because it shows the kind of maths problems ancient Egyptian scribes were interested in and had to deal with, and which were being taught in the schools. The papyrus is divided into three books
Book 1
Division of various numbers of loaves equally between 10 men.
A group of completion calculations involving multiplication of fractions.
Another group of completion calculations involving simple addition of fractions.
Arithmetical solution by trial of equations of the first degree.
Similar equations involving the bushel.
Division of loaves between men in unequal proportions
Book 2
Part 1: Volumes and cubic content in corn.
Cylindrical containers
Rectangular parallelopipedal containers.
Expression in correct form of 1/10, 1/20 up to 1/100 of a bushel, disguised as a sum in cubic content.
Part 2. Areas
Area of square and circle compared.
Rectangle.
Circle.
Triangle.
Truncated Triangle
Trapezoid
Division of given area of land into equal sized fields
Part 3: Batter, or the angle of a slope.
Book 3: Miscellaneous Problems in Arithmetic
Multiplication of fractions
Proportionate values of precious metals.
Division of loaves in unequal proportions.
Division of barley into shares in arithmetical progression.
Division of loaves in unequal proportions.
Daily portion of a yearly ration of fat.
Reckoning of livestock.
Division of 100 bushels of corn in unequal proportions.
So-called pefsu-reckonings. Conversion of grain into bread and beer, and the barter of these last.
Geometrical progression.
Conversion of fractions of the bushel (1/2,1/4, 1/8 etc) in henu.
Food estimate for a poultry yard.
Estimate of food of an ox-stall.
Additions.
Unintelligible group of signs.
Fragment of accounts.
Calendrical entries.
There’s considerable interest in ancient Egypt among Blacks, because it’s been seen since at least the early 19th century as a great Black civilisation. Despite attempts to improve the educational performance of Black children, they still lag behind other ethnic groups like Whites and Asians in schools. I wondered if a way round this would be to try to stimulate their, and other races’ imaginations, with maths problems based on those of the ancient Egyptians. You wouldn’t want to teach them ancient Egyptian mathematical methods, as they’re very different and more convoluted than modern methods and some of them are frankly wrong. But I think you could set kids problems based on the kind of problems budding scribes were taught. You could possibly combine it with Black History month and have the kids dressed up as ancient Egyptians and learn a bit about the civilisation as well.
This is another video from History Debunked. In it, youtuber and author Simon Webb attacks Ethnomatics, sometimes also called Rehumanizing Mathematics. This is a piece of modern pseudo-scholarship designed to help Black children tackle Maths. The idea is that Blacks perform poorly compared at Maths compared to other ethnic groups. This is held to be because Maths is the creation of White men, and this puts Blacks off studying and mastering it.
The solution has been to scrutinise African societies for their indigenous Maths, especially the Dogon of Mali. They have been chosen as the chief model for all this, as they possessed extremely advanced astronomical and mathematical knowledge. In the 1970s there was a book, The Sirius Mystery by Robert K.G. Temple, which claimed that they owed this advance knowledge to contact with space aliens. Apparently this claim was subsequently dropped 10 – 15 years later, and the claim made instead that they were just superlative astronomers and mathematicians themselves. But Dogon Maths is held to be different from White, western Maths because it’s spiritual. History Debunked then goes on to demonstrate the type of pseudo-scientific nonsense this has lead to by providing a link to an Ethnomathematics paper and reading out its conclusion. It’s the kind of pretentious verbiage the late, great Jazzman, Duke Ellington, said stunk up the place. It’s the kind of postmodern twaddle that Sokal and Bricmont exposed in their Intellectual Impostures. It’s deliberately designed to sound impressive without actually meaning anything. There’s a lot of talk about expanding cognitive horizons and possibilities, but History Debunked himself says he doesn’t understand a word of it. And neither, I guess, will most people. Because it doesn’t really mean anything. It’s just there to sound impressive and bamboozle the reader into thinking that somehow they’re thick because they don’t, while the fault is entirely the writers.
I think History Debunked is a man of the right, and certainly his commenters are Conservatives, some with extremely right-wing views. He’s produced a series of videos attacking the pseudo-history being pushed as Black History, and apparently Seattle in America is particularly involved in promoting this nonsense. But he expects it to come over here in a few years. Given the way Black History month has jumped the Atlantic, I think he’s right.
There’s been a particular emphasis on find ancient Black maths and science for some time I know. For a brief while I got on well with a Black studies group when I was a volunteer at the slavery archives in the former Empire and Commonwealth Museum. That was before I read their magazine and got so annoyed with it and its attitude to Whites that I sent them a whole load of material arguing to the contrary, and pointing out that in places like the Sudan, Blacks were being enslaved and oppressed not by White Europeans, but by the Arabs. I also sent them material about the poor Whites of South Africa, who also lived in grinding poverty thanks to Apartheid. This was stuff they really didn’t want to hear, and I was told that if I wanted to talk to them further, I should do so through someone else. They were also interested in finding examples of Black maths and science. I sent them photocopies and notes I’d made of various medieval Muslim mathematicians. These were Arabs and Persians, like al-Khwarizmi, who gave his name to the word algorithm, Omar Khayyam, best known in the west for his Rubayyat, but who was also a brilliant mathematician, al-Haytham, who invented the camera obscura in the 12th century and others, rather than Black. But they were grateful for what I sent them nonetheless, and I thanked me. This was before I blotted my copybook with them.
I’m reposting this piece because, although it comes from the political, it is correct. And you don’t have to be right-wing to recognise and attack this kind of postmodern rubbish. Sokal and Bricmont, the authors of the book I mentioned early attacking postmodernism, were both men of the left. Sokal was a physicist, who taught maths in Nicaragua under the left-wing Sandinista government. They wrote the book because they took seriously George Orwell’s dictum that writing about politics means writing clearly in language everyone can understand. And even if you believe that Black people do need particular help with maths because of issues of race and ethnicity, Ethnomathematics as it stands really doesn’t appear to be it. It just seems to be filling children’s heads with voguish nonsense, rather than real knowledge.
I also remember the wild claims made about the Dogon and their supposed contact with space aliens. Part of it came from the Dogon possessing astronomical knowledge well beyond their level of technology. They knew, for example, that Sirius has a companion star, invisible to the naked eye, Sirius B. They also knew that our solar system had nine planets, although that’s now been subsequently altered. According to the International Astronomical Association or Union or whatever, the solar system has eight planets. Pluto, previously a planet, has been downgraded to dwarf planet, because it’s the same size as some of the planetoids in the Kuiper Belt. Lynn Picknett and Clive Prince discuss this in one their books,The Stargate Conspiracy (London: Little, Brown & Company 1999), which claimed that the American intelligence agencies were secretly preparing a fake UFO landing in order to convince everyone that the space gods really had arrived, and set up a one-world dictatorship. This hasn’t happened, and I’ve seen the Fortean Times and other weird magazines trying to explain their book as a high-level hoax which people took too seriously. I don’t believe this, as they seemed very serious at the time. The Dogon believe that the first human ancestors, and some of their gods, came from the sky. Hence Temple’s claim that they were contacted by space aliens. Picknett and Prince, however, sided with sceptics like Carl Sagan. They argued instead ithat the Dogon owed it to a French priest, anthropologist or colonial administrator, I’ve forgotten which, who visited them in the 1920s and who was extremely interested in astronomy. This seems to me to be far more likely than that they either got it from space aliens or that they far better mathematicians and astronomers than they could have been at their level of development.
The Dogon are fascinating as their homes and villages are laid out to be microcosms of the male and female human body and the universe. The book African Mythology by Geoffrey Parrinder, London: Hamlyn 1967, describes the layout of a Dogon house thus:
The shape of the Dogon house is symbolical. The floor is like the earth and the flat roof like heaven. The vestibule is a man and the central room woman, with store rooms at her sides as arms. The hear at the end is her head. The four posts are the man and woman entwined in union. So the family house represents the unity of man and woman and God and the Earth. This is accompanied by the elevation and ground plan of a typical Dogon house. (p. 49).
There’s also this diagram of an idealised Dogon village:
The caption for the diagrame reads:
Like the house, the Dogon village represents human beings. The smithy is at the head like a hearth in a house. The family houses in the centre and millstones and village represent the sexes. Other altars are the feet. (p. 51).
Truly, a fascinating people and I have no problem anybody wanting to study them. But it should be in anthropology, ethnography or comparative religion, not maths.
But it struck me that if teachers and educators want to enthuse and inspire young minds with what maths Africans were studying, they could start with ancient Egypt and the great Muslim civilisations of the Sahara and north Africa, like Mali. Aminatta Forna in one of her programmes on these civilisations was shown an ancient astronomical text from the medieval library of one of these towns, which she was told showed that Muslims knew the Earth orbited the sun before Copernicus and Galileo. I doubt that very much. It looks like a form of a combined helio-and geocentric system, first proposed by the ancient Greeks, and then taken up by some medieval astronomers not just in Islam, but also in Christian Europe. In this system, all the other planets when round the Sun, which orbited the Earth. Close to the modern system, but not quite. But it showed that the Black citizens of that civilisation were in contact with the great currents of Muslim science, and that they would have had learnt and taught the same kind of Maths that was being investigated and researcher right across the Muslim world, from India to Morocco and further south to Mali. One of the Black educationalists would like to translate one of these books from Arabic, the learned language of Muslim civilisation, and use it as an example of the kind of maths that was also taught in Black Africa.
Or you could go right back to ancient Egypt. Mathematical texts from the Land of the Nile have also survived in the Moscow and Rhind mathematical papyri. These have various maths problems and their solution. For example, problem No. 7 of the Moscow papyrus is about various calculations for a triangle. This runs
Example of calculating a triangle.
If you are told: A triangle of 2 thousands-of-land, the bank of 2 of 2 1/2;
You are to double the area: result 40 (arurae). Take (it) 2 1/2 times; result [100. Take its square root, namely] 10. Evoke 1 from 2 1/2; what results is 2/5. Apply this to 10; result 4. It is 10 (khet) in length by 4 (khet) in breadth. From Henrietta Midonick, The Treasury of Mathematics: 1 (Harmondsworth: Pelican 1965) p. 71.
It’s amazing to think that the boys at the scribal school were being taught all this millennia ago. It gives you a real sense of connection with the ancient schoolkids reading it. You can imagine them, hunched over with their pen and ink, busily cudgeling their brains while the teacher prowls about them. The Babylonians were also renowned as the pioneers of early mathematics. They even uncovered a school when they excavated Ur of the Chaldees in the 1920s, complete with the maths and other texts the schoolboys – female education didn’t exist back then, but I’m willing to be corrected – were required to learn. As a schoolboy character in the Fast Show used to say: ‘Brilliant!’ You don’t need to burden modern African societies like the Dogon with spurious pseudo-history and pseudo-science, when the real historic achievements of ancient Egypt and medieval Africa are so impressive.
It struck me that even if you don’t use the original Egyptian maths texts to teach maths – which would be difficult, as their maths was slightly different. Their method of calculating the area of a field of four unequal sides yields far too high a figure, for example – you could nevertheless inspire children with similar problems. Perhaps you could do it with assistance of a child or two from the class. You could bring them out in front of everyone, give them and ancient Egyptian headdress, and then arranged the lesson so that they helped the teacher, acting as pharaoh, to solve it. Or else pharaoh showed them, his scribes, and thus the class. This is certainly the kind of thing that was done when I was a kid by the awesome Johnny Ball on the children’s maths and science programme, Think of a Number. And every week, as well as showing you a bit of maths and science, he also showed you a trick, which you could find out how to do by dropping him a line. It was the kind of children’s programme that the Beeb did very, very well. It’s a real pity that there no longer is an audience for children’s programmes and their funding has subsequently been cut.
Here’s History Debunked’s video attacking Ethnomathematics. He also attacks a piece of ancient baboon bone carved with notches, which he states has been claimed is an ancient prehistoric African calendar. He provides no evidence in this video to show that it wasn’t, and says its the subject of a later video. If this is the one I’m thinking of, then that is a claim that has been accepted by mainstream archaeologists and historians. See Ivor Grattan-Guinness, The Fontana History of the Mathematical Sciences (London: Fontana Press 1998) p. 24.
If you want to know more about ancient and medieval maths, and that of the world’s many indigenous cultures, see the book Astronomy before the Telescope, edited by Christopher Walker with an introduction by the man of the crumpled suit and monocle himself, Patrick Moore (London: British Museum Press 1998).
This has chapters on astronomy in Europe from prehistory to the Renaissance, but also on astronomy in ancient Egypt, Babylonia, India, Islam, China, Korea and Japan, North and South America, traditional astronomical knowledge in Africa and among Aboriginal Australians, Polynesia and the Maori. It can be a difficult read, as it explores some very technical aspects, but it is a brilliant work by experts in their respective fields.
Henrietta Midonick, The Treasure of Mathematics: 1 (Harmondsworth: Penguin 1968)
I realise that the history of mathematics is an arcane subject, that few people will have much interest in, having struggled enough with the subject at school. But with Black History Month, there is immense interest amongst scholars of Black and Asian history about restoring Black and Asian scientists and mathematicians to their rightful place in history.
I picked up this book in one of the secondhand bookshops in Cheltenham about a year or so ago. It’s a collection of ancient and medieval mathematical texts from Ancient Egypt, Babylon, China, India, Islam, the Jews and, of course, the ancient Greeks. The blurb for it runs
Mathematics is the only true international language. men can communicate more directly, precisely and logically in pure mathematics than in any other tongue. Moreover we have much to learn from the achievements of past civilizations in this field: even modern computers have not fathomed all the intricacies of Stonehenge. In this fascinating collection of original sources (many of them published in a popular edition for the first time) Henrietta Midonick shows individual mathematicians grappling with varied problems – some practical, such as architecture, money valuation, mechanics, astronomy and calendar calculation; others verging on philosophy, such as the existence of zero and the concept of infinity. Her arrangement also demonstrates the growth of key ideas in geometry, arithmetic, logic and calculus.
Volume 1 documents the growth of mathematical science in the civilizations of Babylon, Ancient Egypt, the Mayas, India and China, and assesses the revolutionary discoveries of Plato, Archimedes and Euclid in classical antiquity.
Among the various extracts are pieces on Babylonian mathematics; four geometrical problems from the Moscow Papyrus, which dates from Ancient Egypt, c. 1850 BC; the Rhind Mathematical Papyrus, again from Egypt, c. 1650 BC; the Bakhshali Manuscript, from 4th century AD India; the Mayas – discussing their system of numbers, the calendar, arithmetic and chronology, and the Quipu, the method of keeping statistical records using knots, used by the ancient Incas in South America.
Chinese mathematicians include Wan Wang, from the 12th century BC, Chou Kung, c. 1100 BC; Chang Tsang, died 152; Liu Hui, 3rd century AD; Sun-Tsu, from the same century; Hsia-Hou Yang, 6th century AD; Wang Hs’iao-T’ung, 7th century AD, Li Yeh, c. AD 1178-1265; Ch’in Chiu-Shao, c. AD 1250; Yang Hui, c. AD 1275; Chu Chi-Chieh, c. AD 1300.
The Indian scholars collected include Aryabhata the Elder, c. AD. 476; Brahmagupta, AD 598; and Bhascara Acharya, AD 1114-c. 1185.
It also includes the Algebra of Mohammed ben Musa al-Khowarismi, who founded much of modern algebra, including giving it its modern name.
The two Jewish mathematicians collected include the Mishnat ha-Middot of Rabbi Nehemiah, from c. AD 150; and the Method of Division of Immanuel Ben Jacob Bonfils, c. AD 1350.
The ancient Greeks include Hippocrates of Chios, 5th century BC; an extract from Plato’s Dialogues; the Elements of Euclid of Alexandria, c. 300 BC; Apollonius of Perga’s Conic Sections, from the same period; Archimedes’ On Spirals, Mechanical Problems, and Quadrature of the Parabola, Pappus, c. AD 300, and Proclus, AD 410-485.
Ancient Babylonian Multiplication Table for X 10.
For the non-mathematician like myself these texts aren’t easy reading. There are diagrams to help, but many of them, as the pioneering works of their time, are trying to express difficult mathematical ideas without the modern language of Maths, and so it can be difficult understanding what they are trying to describe. Nevertheless, this is an important collection of some of the classic texts of ancient mathematics on which the structure of modern maths has been built.
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