The easiest question to ask a class is “Any questions so far?”. This rarely produces an answer. Some people tell me that they have a better response with framing this to indicate that questions are expected: “What are your questions?” or “What questions do you have?” for example. Similarly, I’ve given up asking at the end of the course “What do you want me to go over in the revision session?” since some wag usually says “everything”, we all chuckle and that ends the matter. To encourage a response I have found that it is better to ask specific questions, “Are you happy with affine transformations?”, “Do you understand equivalence relations?” The prompt forces them to consider something concrete and it reminds them of what they have been taught. This method can work at any time of the course. “Which exercises are you having trouble with” can be replaced “Do you have trouble with question 6?” The latter forces students to focus. They either do or don’t have a problem with it. I usually pick a question which they will benefit from discussing so even if students are just saying yes to get on with the tutorial at least it will be productive. In summary: Vague general questions are the enemy. Make questions...

## Why is it called a matrix?...

posted by Kevin Houston

Here’s a question for which the answer doesn’t seem to be widely known. Certainly, no one I’ve asked so far has known. They all found the answer interesting though! Why is a matrix called a matrix? If we consult www.etymonline.com for the origins of the word matrix we find matrix (n.) late 14c., “uterus, womb,” from Old French matrice “womb, uterus,” from Latin matrix (genitive matricis) “pregnant animal,” in Late Latin “womb,” also “source, origin,” from mater (genitive matris) “mother” (see mother (n.1)). That doesn’t seem to be much help (but does explain the word matriarchy and why surgeons sometimes refer to the matrix). However, surprisingly “womb” is the origin of the word in mathematics. To see this we go back to JJ Sylvester‘s 1850 article Additions to the articles in the September number of this journal, “On a new class of theorems,” and on Pascal’s theorem in which it is used in this context for the first time. He says For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order. So what is Sylvester talking about here? Well, his interest is in developing what we now think of as Linear Algebra. In particular, he cares about determinants and is thinking of them as arising from a square array of numbers. In the passage above he notes that from an m by n array of numbers...

## Monge and Optimal Transport...

posted by Kevin Houston

Mathematicians are ignorant of their history. They know the names of the greats but generally can repeat only one or two (erroneous) stories about them. For example, if asked for a mathematical story from the French Revolution, then many would plump for a story about Galois, shot at dawn. However, there were many other revolutionary mathematicians. One such is Monge. His name is familiar to differential geometers through Monge form and to analysts of PDEs through the Monge-Ampere equation. Information about him is scant in English. All the important books are in French and so my attempts to study his life in more depth have been thwarted by my lack of fluency. I think I first became interested in him when I was a student. I read a book (the name escapes me) which stated that he had studied the concept of the optimal transport of soil when constructing fortifications. The fact that caught my imagination was that the answer was constrained by the observation that the paths of two particles should not cross. At the time I came up with a counter-example but that was because I didn’t really understand the parameters of the problem. I was greatly interested then when I heard that one of this year’s Hardy Lectures would be about Monge and the optimal transport problem. I was greatly disappointed when I discovered that it clashed with previous commitments. When I was at the conference in Liverpool to celebrate the birthdays of Bill Bruce and Terry Wall Andrew Ranicki told me that the lecture had been recorded. And I’m glad it was, it’s a great talk. It’s given by Etienne Ghys. He takes in the cutting of stones, including how Monge designed a never used plan for the ceiling of...

## The origin of x in maths...

posted by Kevin Houston

I’m still wrestling with the fallout from marking exams. Despite this, I found time to watch a short TED video posted recently that features Terry Moore explaining why x is used as an unknown in mathematics. Watch the video – only 4 minutes – or jump to the spoilers below if you want to know more. The main idea is that the we use x because the Spanish used (chi) as the first sound of the Arabic word for “something” because they couldn’t say the correct “sh” sound. I recollect a similar argument made somewhere else with slightly different details, though, sadly, I cannot remember them precisely, nor their location. I was unable to track down references at the time to verify this argument and so dismissed it as a Just So Story. Since it has resurfaced it would be interesting to see evidence. Does anyone know of any? My reason for asking is that I’ve spent the last few years learning about Greek mathematics and I am interested in how it has been transmitted to us via the Arabic scholars and scientists. So far I’ve only read the popular accounts, Science and Islam: A History by Ehsan Masood and Pathfinders: The Golden Age of Arabic Science by Jim Al-Khalili. The latter is good but spends the first few chapters explaining the history of Islam and various empires. Furthermore, one of the first bits, maybe the first bit, of science to be explained in depth is Eratosthenes’ measurement of the Earth, i.e., a high point of Hellenistic science. Obviously providing a context is important in a book but I still feel as though I don’t know much about the science from the Arabic world between the end of the Greek era and the beginning...

## William Noel at TED on the Archimedes Codex...

posted by Kevin Houston

Regular readers will know that I’m interested in the history of mathematics and am a fan of Archimedes. Well, here’s another video on the Archimedes Codex, this time by William Noel at TED rather than one by his coauthor, Reviel...

## Euclid and Eratosthenes — Greek or African?...

posted by Kevin Houston

Last month the mathematics author John Derbyshire wrote an online article not about mathematics but on his personal views regarding race. Views which eventually got him sacked as a columnist for the publisher. The Guardian newspaper responded through an article by Jonathan Farley. My post today is not about race but rather about some points made in the comments to Farley’s article. In his article Farley said … Euclid, Eratosthenes and other African mathematicians outshone Europe’s brightest stars for millennia. In the comments section it was asked They are known as Greek mathematicians. Why are they quoted in an article about Black mathematicians? Now, one should avoid getting involved in fights in comments section and fortunately someone had replied in a comment later highlighted by Guardian staff Well, Euclid is ‘Euclid of Alexandria’ which is in Egypt of course and Eratosthenes was born in Cyrene (modern Libya). So I don’t think its erroneous to say they were black mathematicians. Various arguments were made later in the comments about whether North African counted as black. I’m not going to get into this argument either. Instead my post is about the precise origins of Euclid and Eratosthenes. In my geometry and history of mathematics courses I tell my students that when we talk of Greek mathematicians, we should not think of them as swanning around in Athens dressed in togas. Instead they came from all over the Mediterranean, from what we now call Italy, Egypt, Turkey, Libya and so on. Some even came from Greece. (And they didn’t wear togas. That was the Romans. The Greeks wore a chiton, a type of tunic.) A good example of this is the greatest Greek scientist, Archimedes, who was from Syracuse in Sicily. But what about Euclid and Eratosthenes?...