I expect you've heard the ancient joke about the cross-eyed teacher. He couldn't control his pupils, you see. 

I always assumed the two different meanings of pupil evolved independently and their being homonyms is pure accident. But not so! While looking something else up, I found to my astonishment (whatever could the connection be?) they have the same root and their origin is a fascination story.

In Latin there was a word pupus, 'boy' and a feminine form pupa, 'girl'. These had diminutives, pupillus and pupilla, 'little boy' and 'little girl'. These words were used to describe orphans, or young persons under the tutelage of an adult. In French this became pupille and was borrowed into English to simply mean a young student. (A vestige of the tutelage meaning remains when pupil is used to describe a trainee barrister.) 

Pupilla also meany 'doll' and this is what the pupil of the eye gets its name from.

If you look into someone's eyes you may see a tiny reflection in their pupils. Light passing through the pupil is absorbed, making the pupil look black, but some light is reflected off its surface and in favourable circumstances you may see a tiny reflected image of yourself - a 'doll'! and pupilla was the name given to this doll.

Fascinatingly Ancient Greek had a word κόρη (kori), 'little girl', and this too was used to mean the pupil of the eye.

This little doll in the eye really exists, although I think you would need a special camera to photograph it, but here is rather nice photo by Vladimer Shioshvili

'Stephanie was looking out through the window, looking at clouds passing...'

And now you can probably make a guess about where puppet and puppy come from.

Picture credit: Vladimer Shioshvili, 17 October 2007, licensed under Creative Common.

My bother sent me this picture of a pair of Great Crested Grebes (Podiceps cristatus) with their chicks, which you can see riding on a parent's back.

The RSPB describes the bird like this

'a delightful, elegant waterbird with decorative head plumes'

About

Courtship

These birds are famous for their courtship ritual, often described as a dance.

A floating nest

Their nest is a floating platform of reeds and other vegetation.

In at the Deep End

The chicks ride on a parent's and when the parent dives down they go too! But they take to it like ... Grebes to water.

Victorian Ladies' Headgear

Their head plumes were an object of fashion in the 19th century, which brought the species to the brink of extinction.

The name

From French grèbe, so much is certain. Possibly this is a borrowing into French from Breton, related to Welsh crib, 'crest', including of a mountain ridge. or 'comb' as in a cock's comb. But this is not universally accepted.

This is a solution to the problem I posted 16 May 2026.

In the diagram triangles ABC and CDE are equilateral, with points A, C and E lying on a straight line. The problem is to prove CP and CQ have the same length.

There are probably many proofs - for example using coordinate geometry or complex number - but here is a short one using Euclidean geometry.

In the second diagram the coloured triangles ACD and BCE are congruent ('two sides and the included angle'), because AC = BC, CD = CE, and angle ACD = 120° = angle BCE . The two angles marked are therefore equal.

In the third diagram the coloured triangle CPD and the shaded triangle CQE are congruent ('two angles and the included side'), because angle PCD = 60° = angle QCE, angle PDC = = angle QEC and side CD = side CE.

Consequently CP = CQ which was to be shown.

These attractive little rodents are found all across Europe and Asia, as far as China. In the British Isles they seem to be mainly aquatic (as suggested by the species name amphibius) but on the Continent the same animal is a land vole (which is why Linnaeus named the genus Avicola = 'field dweller).

Across its range it is not at risk but in Britain it has declined and is a protected species. The main reasons for decline seem to be habitat loss and predation by American mink, descended from animal who escaped or were released from fur farms.

However conservation efforts seem to have made a difference and the population seems to have picked up to some extent, but they are still much less common than they pnce were.

I knew squirrel is from Ancient Greek skiouros (σκίουρος), ‘shadow tail’. To me this always seemed quite apt; a squirrel’s bushy tail does resemble a shadow. (Although apparently Ancient Greeks explained it as being because a squirrel carries its own portable sunshade.) 

The elements of skiourus are ski-, ‘shadow’, and ourus, which is the same word as ‘arse’.

Yes, really. I was staggered recently when I learn this but there is good evidence that Proto Indo-European (PIE), the ancestor of English, Greek, and many other languages, had a word *ors-, ‘backside’.

By now you may have spotted that this is the second element in cynosure, which today means focus of attention; or something or someone to which all eyes turn. The word is derived from Greek kunosoura (κυνόσουρα), “dog's tail”, first element kuon (κύων), ‘dog’ and the -oura element is a variant form of ouros.

Kunosoura was the Greek name for the constellation we now call Ursa Minor.

‘All eyes were on it’ because Greek sailors looked to it to find North. My sketch shows things as they are today; but because the Earth’s axis wobbles, the star that was nearest to North in classical Greek times was the one I’ve ringed and that is what Greek navigators used.

And now... the wheatear (Oenanthe Oenanthe). This bird when seen in flight has a very eye-catching white rump.

The general consensus is that it was originally called whiterse, ‘white-arse’, from this striking feature, but the word morphed into wheatear. This might be because the original sense got forgotten and the compound of familiar words wheat + ear was something speakers felt comfortable with. Or there might have been an element of prudery; perhaps arse was just too ‘Anglo-Saxon’ but ear did not offend.

This ‘explaining’ a word’s origin as a compound of more familiar and homely terms is called folk etymology. A nice example is sparrowgrass, a green vegetable better known today as asparagus, but often called sparrowgrass historically and in some dialects to this day.

And now back to our squirrel. Well-respected scholars have been sceptical about the etymology I began this post with. They suggest instead that it was a borrowing from a pre-Greek language and the Greeks rationalised as skiouros by folk etymology. This could well be true, but we shall never know, and the idea of an animal that carries its own beach umbrella around with it is too appealing to give up lightly.

Covering problems, which ask how a shape can be covered with other shapes, are part of what's called combinatorial mathematics. They often appear in recreational mathematics. have applications to real-world problems such as siting mobile phone mast to get adequate coverage, and are a subject of active current research.

Covering problems are often easy to state but even in simple cases the answers can be difficult to establish, because when you are arranging a bunch of shapes it's hard to be sure all the possibilities have been thought of.

One I thought of the other day and posted in this blog is 

What is the smallest equilateral triangle that can cover every triangle whose longest side has length 1?

This is about as simple as it gets but it's not trivial. The first idea you might have, an equilateral triangle with sides of length 1, turns out not to be the answer; a bigger triangle is needed.

I haven't proved to my satisfaction what the smallest possible answer is but I can prove the following.

Any triangle whose longest side is 1 can be covered by an equilateral triangle of side length .

To see this consider Figures 1 and 2 below.

In Figure 1 AB is the longest side of the triangle we wish to cover, so its length is 1. Where can the third vertex of the triangle, call it C, be located?

If we draw circles of radius 1 centered at A and B then C must be in the lens-shaped region AXBY; if not, C would be more than 1 away from at least one of A and B , contradicting AB being the longest side.

From symmetry it is enough to just consider the shaded sector in Figure 1. In Figure 2 we see this sector is covered by equilateral triangle AX₁B_1, which therefore covers all three vertices of the triangle we want to cover and thus covers the whole of that triangle. 

The side length of AX₁B₁ is , which completes the proof.

What is the smallest equilateral triangle that can be guaranteed to cover any triangle whose longest side has length ? We might be tempted to think an equilateral triangle whose side length is also will do the job, as in (1) below.

However if a base angle is just over as shown in (2) the side marked is still the longest side but now the triangle we want to cover cannot fit into the equilateral triangle. Moving the equilateral triangle cannot help; the only way to cover two points that are apart is if they lie at vertices of the equilateral triangle and the same problem will arise whatever pair we pick.

Can you work out how large the equilateral triangle has to be before we can be confident it can cover any triangle whose longest side is ?

It's one of my favourite flower names. It's from Old French pensee, 'thought', but already in French applied to the flower, with the sense of a remembrance. Pensee comes from Latin pensare, 'weigh up' and this is where it gets really interesting, because pensare comes from a word pendere connected with weighing or hanging something up.

This in turn goes back to a Proto Indo-European root *(s)pen- which is the ultimate origin of a long list of words, some quite unexpected, such as as dispensary, expensive, peso, penthouse, spider and spontaneous.

I'll finish with a rather touching little poem ca. 1450 I found in the Middle English Compendium.

The lynyng of hit was with nedille wrought..With litille, litille flowris soft, The soven and the daisy, But most of pancy.

Soven must be connected with souvenance (think souvenir) so presumably another flower symbolising remembrance, but I couldn't find what it is. Perhaps we can never know.

I don't know why but I've always had a soft spot for the vocative. I wish English still had it. What is it? Basically a special form of a name you use when addressing someone or something.

A really nice and surprising example, if you didn't know about it, is Scottish Gaelic Seumas, 'James', which becomes the vocative A Sheumais when you directly address someone of that name. This is pronounced uh-haymish and now you know where the name Hamish comes from.

I don't know any Gaelic but I know some Greek. In Greek, when speaking directly to someone you use the vocative. For some names the end of the name may change, for example if you address Kostas it will be as Kosta, Yiannis as Yianni. Other names may not change their spelling but are still regarded as vocatives; for example Maria or Anna, and I think that speakers, even non-native ones,  sense them as vocatives

This can be confusing to people learning Greek, who when hearing someone address Kostas, using the vocative Kosta of course, tend to assume he is called Kosta when he's actually called Kostas (hope you're keeping up!) Even more confusing, if you speak about Kostas, you must refer to him as 'The Kostas', Ο Κοστας.

The vocative is also used when addressing someone by a title-based form. For example 'Doctor' in Greek is Yiatros but when I address the doctor I have to say Yiatre. I could even speak to my dog (skylos) in the vocative, Kaló skyle!, 'Good dog'.

Originally all the branches of the large Indo-European family used word endings to mark what role a noun played in a sentence, for example being the object of an action ('I patted the dog')'or being a possessor ('the dog's dinner) or a recipient ('I gave the dog a bone'). There were eight different 'cases' altogether, including our vocative. 

Over time some branches of the Indo European family have eroded or abolished the case system in favour of things like word order, and nowadays cases have largely disappeared from all the Germanic languages bar Icelandic and German itself, and all the Romance languages bar Romanian. This included the poor vocative, now only hanging on in Icelandic and Romanian respectively. 

But it's alive and well in Greek, as we have seen and in many other branches as well. An interesting exception is Russian, which unlike most Slavic languages lacks a vocative. It was abolished by the Russian government in 1918 on the grounds that it was archaic and little used. However, and this is fascinating, an informal 'neo-vocative' has apparently emerged in Modern Russian. for example Sasha would become Sash.

In the world of triangle we find a rich etymological heritage, with most words having old roots, with many going back to Proto-Indo-European (PIE), the 5000-ish year old ancestor of most European languages and many from Northern India and the Iranian region.

Triangle is from Old French, from Latin triangulum. The tri- is from PIE *treyes, 'three' obviously, but -angle is less obvious; the PIE root is *ang or *ank, 'bend', which is also the source of ankle.

It also seems the Angles (as in East Anglia) took their name from the same root, perhap in reference to a bend in the coastline of their original homeland, or alternatively because the hooks they used to catch fish, so they would be 'The people of the fishing-hook' according to this theory.

Triangles come in many flavours and I've sketched some below. Triangle can be classified by the size of their largest angle, or by how many equal sides they have. In this post I'm only looking at the first classification, by angle, and I'll look at classification by equal sides in another post.

Number 1, the Acute, has all angles less than 90 degrees, so they are 'sharp'. Acute is derived from Latin acutis 'sharp', from PIE *ak-, 'sharp' or 'pointy', from which we get many words, such as acid, acropolis, acupuncture, eager and oxygen.

Number 2, the Obtuse, has an angle greater than 90 degrees. Obtuse means 'not sharp', from Latin obtusus, 'blunt', from PIE *(s)teu-,'beat' or 'push. This is the source also of type (from being struck), stupid (as in struck insensible), student (students push forward keenly) and stupendous (stunning).

Number 3, the Right-Angled, has a 90 degree angle, and sits right (!) on the cusp between (1) and (2). The right part is a translation of Latin rectus, 'upright', into Old English riht, 'straight', which both descends ultimately from PIE *reg-, the root of a whole host of words around being correct or regulated, or reigning, or being a maharajah, or being reckless, or being regal.

And it is the first element of my given name Richard!This is a so-called dithemic name, a compound of two Germanic elements meaning 'strong ruler' but with the ruler element first. So my name has the same PIE origin as right as in right angle.

Finally an old joke. See if you can spot how this ties in with PIE *reg-,

Did you hear about the king who was only 12 inches tall? He was a lousy king but a great ruler.

From a friend who knows my love of wild flowers.

The scientific name derives from Hyakinthos, a mythical Spartan prince who was a lover of Apollo. One day in a sporting apollo accidentally killed Hyakinthos with a discus. This (or some other, its not certain what flower was meant) plant was said to have sprung from the blood of the unfortunate youth.

The name hyakinthos (ὑάκινθος) is etymologically interesting. Greek words with -nth- in them, such as plinth or Corinth have an ancient pre-Greek and pre-Indo European origin, so I suppose the name hyacinth harks back to the language of a Bronze Age Mediterranean people.

Here's a photo my brother Simon took yesterday of a Grey Heron's nest with chicks.

These imposing birds can be up to 1 m tall, with a wing span of up to 2 m.