## The Special Beards of Relativity

Posted in Beards, History, The Universe and Stuff with tags , , , , , on December 7, 2022 by telescoper

I’ve recently moved on to the part about Special Relativity in my module on Mechanics and Special Relativity and this afternoon I’m going to talk about the Lorentz-Fitzgerald contraction or, as it’s properly called here in Ireland, the Fitzgerald-Lorentz contraction.

The first thing to point out is that the physicists George Francis Fitzgerald and Hendrik Lorentz, though of different nationality (the former Irish, the latter Dutch), both had fine beards:

One of the interesting things you find if you read about the history of physics just before Albert Einstein introduced his theory of special relativity in 1905 was how many people seemed to be on the verge of getting the idea around about the same time. Fitzgerald and Lorentz were two who were almost there; Poincaré was another. It was as if special relativity was `in the air’ at the time. It did, however, take a special genius like Einstein to crystallize all that thinking into a definite theory.

Special relativity is fun to teach, not least because it throws up interesting yet informative paradoxes (i.e. apparent logical contradictions) arising from  that you can use to start a discussion. They’re not really logical contradictions, of course. They just challenge `common sense’ notions, which is a good thing to do to get people thinking.

Anyway, I thought I’d mention one of my favorite such paradoxes arising from a simple Gedankenerfahrung (thought experiment) here.

Imagine you are in a railway carriage moving along a track at constant speed relative to the track. The carriage is dark, but at the centre of the carriage is a flash bulb. At one end (say the front) of the carriage is a portrait of Lorentz and at the other (say the back) a portrait of Fitzgerald; the pictures are equidistant from the bulb and next to each portrait is a clock.The two clocks are synchronized in the rest frame of the carriage.

At a particular time the flash bulb goes off, illuminating both portraits and both clocks for an instant.

It is an essential postulate of special relativity that the speed of light is the same to observers in any inertial frame, so that an observer at rest in the centre of the carriage sees both portraits illuminated simultaneously as indicated by the adjacent clocks. This is because the symmetry of the situation means that light has to travel the same distance to each portrait and back.

Now suppose we view the action from the point of view of a different inertial observer, at rest by the trackside rather than on the train, who is positioned right next to the centre of the carriage as the flash goes off. The light flash travels with the same speed in the second observer’s frame, but this observer sees* the back of the carriage moving towards the light signal and the front moving away. The result is therefore that this observer sees the two portraits light up at different times. In this case the portrait of Fitzgerald is lit up before the portrait of Lorentz.

Had the train been going in the opposite direction, Lorentz would have appeared before Fitzgerald. That just shows that whether its Lorentz-Fitzgerald contraction or Fitzgerald-Lorentz contraction is just a matter of your frame of reference…

But that’s not the paradoxical thing. The paradox is although the two portraits appear at different times to the trackside observer, the clocks nevertheless display the same time….

*You have to use your imagination a bit here, as the train has to be travelling at a decent fraction of the speed of light. It’s certainly not an Irish train.

## Lorentz-Fitzgerald or Fitzgerald-Lorentz?

Posted in Beards, History, The Universe and Stuff with tags , , , , on December 9, 2020 by telescoper

I’ve recently moved on to the part about Special Relativity in my module on Mechanics and Special Relativity and this afternoon I’m going to talk about the Lorentz-Fitzgerald contraction or, as it’s properly called here in Ireland, the Fitzgerald-Lorentz contraction.

The first thing to point out is that the physicists George Francis Fitzgerald and Hendrik Lorentz, though of different nationality (the former Irish, the latter Dutch), both had fine beards:

George Francis Fitzgerald (1851-1901)

Hendrik Lorentz (1853-1928)

One of the interesting things you find if you read about the history of physics just before Albert Einstein introduced his theory of special relativity in 1905 was how many people seemed to be on the verge of getting the idea around about the same time. Fitzgerald and Lorentz were two who were almost there; Poincaré was another. It was as if special relativity was `in the air’ at the time. It did, however, take a special genius like Einstein to crystallize all that thinking into a definite theory.

Special relativity is fun to teach, not least because it throws up interesting yet informative paradoxes (i.e. apparent logical contradictions) arising from  that you can use to start a discussion. They’re not really logical contradictions, of course. They just challenge `common sense’ notions, which is a good thing to do to get people thinking.

Anyway, I thought I’d mention one of my favorite such paradoxes arising from a simple Gedankenerfahrung (thought experiment) here.

Imagine you are in a railway carriage moving along a track at constant speed relative to the track. The carriage is dark, but at the centre of the carriage is a flash bulb. At one end (say the front) of the carriage is a portrait of Lorentz and at the other (say the back) a portrait of Fitzgerald; the pictures are equidistant from the bulb and next to each portrait is a clock.The two clocks are synchronized in the rest frame of the carriage.

At a particular time the flash bulb goes off, illuminating both portraits and both clocks for an instant.

It is an essential postulate of special relativity that the speed of light is the same to observers in any inertial frame, so that an observer at rest in the centre of the carriage sees both portraits illuminated simultaneously as indicated by the adjacent clocks. This is because the symmetry of the situation means that light has to travel the same distance to each portrait and back.

Now suppose we view the action from the point of view of a different inertial observer, at rest by the trackside rather than on the train, who is positioned right next to the centre of the carriage as the flash goes off. The light flash travels with the same speed in the second observer’s frame, but this observer sees* the back of the carriage moving towards the light signal and the front moving away. The result is therefore that this observer sees the two portraits light up at different times. In this case the portrait of Fitzgerald is lit up before the portrait of Lorentz.

Had the train been going in the opposite direction, Lorentz would have appeared before Fitzgerald. That just shows that whether its Lorentz-Fitzgerald contraction or Fitzgerald-Lorentz contraction is just a matter of your frame of reference…

But that’s not the paradoxical thing. The paradox is although the two portraits appear at different times to the trackside observer, the clocks still appear show the same time….

*You have to use your imagination a bit here, as the train has to be travelling at a decent fraction of the speed of light. It’s certainly not an Irish train.

## Clifford’s `Space-Theory of Matter’

Posted in Beards, History, mathematics, The Universe and Stuff with tags , , , on February 26, 2020 by telescoper

Well, here’s another thing I didn’t know until I was informed by Twitter.

Way back in 1876 –  forty years before Einstein presented his Theory of General Relativity – the mathematician W.K. Clifford (who is most famous nowadays for the Clifford Algebra) presented a short paper in the Proceedings of the Cambridge Philosophical Society in which he speculated that space might be described by Riemannian rather than Euclidean Geometry.

Here are a couple of excerpts:

and

The paper does not contain any actual equations, and his concentration on small scales rather than large was misguided, but it is quite remarkable that he was thinking about such matters such a long time ago!

Unfortunately Clifford died very young, in 1879, at the age of 33, tuberculosis. Had he lived longer he might have been able to develop these ideas a bit further.

As a postscript I should mention that Clifford had an impressive beard.

## The Eddington Eclipse Expeditions and Astronomy Ireland

Posted in History, Talks and Reviews, The Universe and Stuff with tags , , , , , , on December 10, 2019 by telescoper

After a full shift during the day at Maynooth University, yesterday evening I made my way into Dublin to give a talk to a very large audience in the famous Schrödinger Lecture Theatre in Trinity College, Dublin, an event organized by Astronomy Ireland. I have given a number of talks on the topic of the 1919 Eclipse Expeditions during this centenary year, but I think this one had the biggest audience! We adjourned to a local pub for a drink afterwards before I dashed off to get the last train back to Maynooth.

Here are the slides I used during the talk:

This time there was an important addition to my usual talk, courtesy of Professor Peter Gallagher of DIAS. He brought along the actual 4″ object glass used in the expedition to Sobral (Brazil) in 1919. I have previously only shown a picture of it. The appearance of the actual lens drew a spontaneous round of applause from the audience, and I have to admit it was a remarkable feeling to hold a little piece of history in my hand!

Obviously I was careful not to drop this item. It is on permanent display in Dunsink Observatory, by the way, if you want to see it yourself. I hope it made its way back here safely!

After the talk was over I was chatting to a couple of members of the audience when Peter Gallagher took this nice picture actually through the lens:

Picture Credit: Peter Gallagher

I look rather old in this picture. Obviously a trick of the lens.

## The Funeral of Lorentz

Posted in History, The Universe and Stuff with tags , , on December 6, 2019 by telescoper

In a post a couple of days ago I mentioned the Dutch physicist Hendrik Lorentz, whose work helped establish the foundations of the theory of special relativity.

Hendrik Lorentz (1853-1928)

Doing a quick google about Lorentz I came across this remarkable silent footage of his funeral in 1928 in the town of Haarlem in the Netherlands.

The funeral took place at Haarlem at noon on Friday, February 10. At the stroke of twelve the State telegraph and telephone services of Holland were suspended for three minutes as a revered tribute to the greatest man the Netherlands has produced in our time. It was attended by many colleagues and distinguished physicists from foreign countries. The President, Sir Ernest Rutherford, represented the Royal Society and made an appreciative oration by the graveside.

The footage of the funeral procession shows a lead carriage followed by ten mourners, followed by a carriage with the coffin, followed in turn by at least four more carriages, passing by a crowd at the Grote Markt, Haarlem from the Zijlstraat to the Smedestraat, and then back again through the Grote Houtstraat towards the Barteljorisstraat, on the way to the “Algemene Begraafplaats” at the Kleverlaan (northern Haarlem cemetery).
Einstein later gave a eulogy at a memorial service at Leiden University.

It was clearly a very grand affair which demonstrates high regard in which Lorentz was held not only by physicists but by the wider public.

## The Relativity of Beards

Posted in Beards, History, The Universe and Stuff with tags , , , , on December 4, 2019 by telescoper

In my first-year module on Mechanics and Special Relativity, I’ve just moved on to the part about Special Relativity and this afternoon I’m going to talk about the Lorentz-Fitzgerald contraction or, as it’s properly called here in Ireland, the Fitzgerald-Lorentz contraction.

The first thing to point out is that the physicists George Francis Fitzgerald and Hendrik Lorentz, though of different nationality (the former Irish, the latter Dutch), both had fine beards:

George Francis Fitzgerald (1851-1901)

Hendrik Lorentz (1853-1928)

One of the interesting things you find if you read about the history of physics just before Albert Einstein introduced his theory of special relativity in 1905 was how many people seemed to be on the verge of getting the idea around about the same time. Fitzgerald and Lorentz were two were almost there; Poincaré was another. It was like special relativity was `in the air’ at the time. It did, however, take a special genius like Einstein to crystallize all that thinking into a definite theory.

Special relativity is fun to teach, not least because it throws up interesting yet informative paradoxes (i.e. apparent logical contradictions) arising from  that you can use to start a discussion. They’re not actually paradoxes really logical contradictions, of course. They just challenge `common sense’ notions, which is a good thing to do to get people thinking.

Anyway, I thought I’d mention one of my favorite such paradoxes arising from a simple Gedankenerfahrung (thought experiment) here.

Imagine you are in a railway carriage moving along a track at constant speed relative to the track. The carriage is dark, but at the centre of the carriage is a flash bulb. At one end (say the front) of the carriage is a portrait of Lorentz and at the other (say the back) a portrait of Fitzgerald; the pictures are equidistant from the bulb and next to each portrait is a clock.The two clocks are synchronized in the rest frame of the carriage.

At a particular time the flash bulb goes off, illuminating both portraits and both clocks for an instant.

It is an essential postulate of special relativity that the speed of light is the same to observers in any inertial frame, so that an observer at rest in the centre of the carriage sees both portraits illuminated simultaneously as indicated by the adjacent clocks. This is because the symmetry of the situation means that light has to travel the same distance to each portrait and back.

Now suppose we view the action from the point of view of a different inertial observer, at rest by the trackside rather than on the train, who is positioned right next to the centre of the carriage as the flash goes off. The flight flash travels with the same speed in the second observer’s frame, but this observer sees* the back of the carriage moving towards the light signal and the front moving away. The result is therefore that this observer sees the two portraits light up at different times. In this case the portrait of Fitzgerald is lit up before the portrait of Lorentz.

Had the train been going in the opposite direction, Lorentz would have appeared before Fitzgerald. That just shows that whether its Lorentz-Fitzgerald contraction or Fitzgerald-Lorentz contraction is just a matter of your frame of reference…

But that’s not the paradoxical thing. The paradox is although the two portraits appear at different times to the trackside observer, the clocks still appear show the same time….

*You have to use your imagination a bit here, as the train has to be travelling at a decent fraction of the speed of light. It’s certainly not an Irish train.

## Lights all askew in the Heavens – the 1919 Eclipse Expeditions (Updated)

Posted in History, Talks and Reviews, The Universe and Stuff with tags , , , , , on June 3, 2019 by telescoper

Here is a video of my talk at the Open Meeting of the Royal Astronomical Society on April 12 2019. Was it really so long ago?

You can find the slides here:

## The 1919 Eclipse: That Was The Talk That Was…

Posted in History, The Universe and Stuff with tags , , , , , on May 29, 2019 by telescoper

Well, I did my talk this afternoon to mark the centenary of the 1919 Eclipse Experiment that was performed on May 29th 1919. It’s a good job we changed the venue to a bigger lecture theatre than originally booked because even the new one was full! Thanks to everyone who came, and I hope you enjoyed the talk!

Anyway, here are the slides if you’d like to see them:

Here is a picture of me about to start:

Now that the centenary has passed I promise to post a bit less about this topic, although there are still a few things coming up that I might mention…

## The Centenary of the 1919 Eclipse Expeditions

Posted in History, The Universe and Stuff with tags , , , , on May 29, 2019 by telescoper

Well, the big day has arrived. Today, 29th May 2019, is the centenary of the 1919 Solar Eclipse during which an experiment was carried out to test Einstein’s theory of general relativity. I’m giving a public talk this afternoon and will post the slides afterwards.

In the meantime, however, I’ll just re-post his little piece which is based on an article I wrote some years ago for Firstscience.

–0–

The Eclipse that Changed the Universe

A total eclipse of the Sun is a moment of magic: a scant few minutes when our perceptions of the whole Universe are turned on their heads. The Sun’s blinding disc is replaced by ghostly pale tentacles surrounding a black heart – an eerie experience witnessed by hundreds of millions of people throughout Europe and the Near East last August.

But one particular eclipse of the Sun, eighty years ago, challenged not only people’s emotional world. It was set to turn the science of the Universe on its head. For over two centuries, scientists had believed Sir Isaac Newton’s view of the Universe. Now his ideas had been challenged by a young German-Swiss scientist, called Albert Einstein. The showdown – Newton vs Einstein – would be the total eclipse of 29 May 1919.

Newton’s position was set out in his monumental Philosophiae Naturalis Principia Mathematica, published in 1687. The Principia – as it’s familiarly known – laid down a set of mathematical laws that described all forms of motion in the Universe. These rules applied as much to the motion of planets around the Sun as to more mundane objects like apples falling from trees.

At the heart of Newton’s concept of the Universe were his ideas about space and time. Space was inflexible, laid out in a way that had been described by the ancient Greek mathematician Euclid in his laws of geometry. To Newton, space was the immovable and unyielding stage on which bodies acted out their motions. Time was also absolute, ticking away inexorably at the same rate for everyone in the Universe.

Sir Isaac Newton, painted by Sir Godfrey Kneller. Picture Credit: National Portrait Gallery,

For over 200 years, scientists saw the Cosmos through Newton’s eyes. It was a vast clockwork machine, evolving by predetermined rules through regular space, against the beat of an absolute clock. This edifice totally dominated scientific thought, until it was challenged by Albert Einstein.

In 1905, Einstein dispensed with Newton’s absolute nature of space and time. Although born in Germany, during this period of his life he was working as a patent clerk in Berne, Switzerland. He encapsulated his new ideas on motion, space and time in his special theory of relativity. But it took another ten years for Einstein to work out the full consequences of his ideas, including gravity. The general theory of relativity, first aired in 1915, was as complete a description of motion as Newton had prescribed in his Principia. But Einstein’s description of gravity required space to be curved. Whereas for Newton space was an inflexible backdrop, for Einstein it had to bend and flex near massive bodies. This warping of space, in turn, would be responsible for guiding objects such as planets along their orbits.

Albert Einstein (left), pictured with Arthur Stanley Eddington (right). Picture Credit: Royal Greenwich Observatory.

By the time he developed his general theory, Einstein was back in Germany, working in Berlin. But a copy of his general theory of relativity was soon smuggled through war-torn Europe to Cambridge. There it was read by Arthur Stanley Eddington, Britain’s leading astrophysicist. Eddington realised that Einstein’s theory could be tested. If space really was distorted by gravity, then light passing through it would not travel in a straight line, but would follow a curved path. The stronger the force of gravity, the more the light would be bent. The bending would be largest for light passing very close to a very massive body, such as the Sun.

Unfortunately, the most massive objects known to astronomers at the time were also very bright. This was before black holes were seriously considered, and stars provided the strongest gravitational fields known. The Sun was particularly useful, being a star right on our doorstep. But it is impossible to see how the light from faint background stars might be bent by the Sun’s gravity, because the Sun’s light is so bright it completely swamps the light from objects beyond it.

A scientific sketch of the path of totality for the 1919 eclipse. Picture Credit: Royal Greenwich Observatory.

Eddington realised the solution. Observe during a total eclipse, when the Sun’s light is blotted out for a few minutes, and you can see distant stars that appear close to the Sun in the sky. If Einstein was right, the Sun’s gravity would shift these stars to slightly different positions, compared to where they are seen in the night sky at other times of the year when the Sun far away from them. The closer the star appears to the Sun during totality, the bigger the shift would be.

Eddington began to put pressure on the British scientific establishment to organise an experiment. The Astronomer Royal of the time, Sir Frank Watson Dyson, realised that the 1919 eclipse was ideal. Not only was totality unusually long (around six minutes, compared with the two minutes we experienced in 1999) but during totality the Sun would be right in front of the Hyades, a cluster of bright stars.

But at this point the story took a twist. Eddington was a Quaker and, as such, a pacifist. In 1917, after disastrous losses during the Somme offensive, the British government introduced conscription to the armed forces. Eddington refused the draft and was threatened with imprisonment. In the end, Dyson’s intervention was crucial persuading the government to spare Eddington. His conscription was postponed under the condition that, if the war had finished by 1919, Eddington himself would lead an expedition to measure the bending of light by the Sun. The rest, as they say, is history.

The path of totality of the 1919 eclipse passed from northern Brazil, across the Atlantic Ocean to West Africa. In case of bad weather (amongst other reasons) two expeditions were organised: one to Sobral, in Brazil, and the other to the island of Principe, in the Gulf of Guinea close to the West African coast. Eddington himself went to Principe; the expedition to Sobral was led by Andrew Crommelin from the Royal Observatory at Greenwich.

British scientists in the field at their observing site in Sobral in 1919. Picture Credit: Royal Greenwich Observatory

The expeditions did not go entirely according to plan. When the day of the eclipse (29 May) dawned on Principe, Eddington was greeted with a thunderstorm and torrential rain. By mid-afternoon the skies had partly cleared and he took some pictures through cloud.

Meanwhile, at Sobral, Crommelin had much better weather – but he had made serious errors in setting up his equipment. He focused his main telescope the night before the eclipse, but did not allow for the distortions that would take place as the temperature climbed during the day. Luckily, he had taken a backup telescope along, and this in the end provided the best results of all.

After the eclipse, Eddington himself carefully measured the positions of the stars that appeared near the Sun’s eclipsed image, on the photographic plates exposed at both Sobral and Principe. He then compared them with reference positions taken previously when the Hyades were visible in the night sky. The measurements had to be incredibly accurate, not only because the expected deflections were small. The images of the stars were also quite blurred, because of problems with the telescopes and because they were seen through the light of the Sun’s glowing atmosphere, the solar corona.

Before long the results were ready. Britain’s premier scientific body, the Royal Society, called a special meeting in London on 6 November. Dyson, as Astronomer Royal took the floor, and announced that the measurements did not support Newton’s long-accepted theory of gravity. Instead, they agreed with the predictions of Einstein’s new theory.

The final proof: the small red line shows how far the position of the star has been shifted by the Sun’s gravity. Each star experiences a tiny deflection, but averaged over many exposures the results definitely support Einstein’s theory. Picture Credit: Royal Greenwich Observatory.

The press reaction was extraordinary. Einstein was immediately propelled onto the front pages of the world’s media and, almost overnight, became a household name. There was more to this than purely the scientific content of his theory. After years of war, the public embraced a moment that moved mankind from the horrors of destruction to the sublimity of the human mind laying bare the secrets of the Cosmos. The two pacifists in the limelight – the British Eddington and the German-born Einstein – were particularly pleased at the reconciliation between their nations brought about by the results.

But the popular perception of the eclipse results differed quite significantly from the way they were viewed in the scientific establishment. Physicists of the day were justifiably cautious. Eddington had needed to make significant corrections to some of the measurements, for various technical reasons, and in the end decided to leave some of the Sobral data out of the calculation entirely. Many scientists were suspicious that he had cooked the books. Although the suspicion lingered for years in some quarters, in the end the results were confirmed at eclipse after eclipse with higher and higher precision.

In this cosmic ‘gravitational lens,’ a huge cluster of galaxies distorts the light from more distant galaxies into a pattern of giant arcs. Picture Credit: NASA

Nowadays astronomers are so confident of Einstein’s theory that they rely on the bending of light by gravity to make telescopes almost as big as the Universe. When the conditions are right, gravity can shift an object’s position by far more than a microscopic amount. The ideal situation is when we look far out into space, and centre our view not on an individual star like the Sun, but on a cluster of hundreds of galaxies – with a total mass of perhaps 100 million million suns. The space-curvature of this immense ‘gravitational lens’ can gather the light from more remote objects, and focus them into brilliant curved arcs in the sky. From the size of the arcs, astronomers can ‘weigh’ the cluster of galaxies.

Einstein didn’t live long enough to see through a gravitational lens, but if he had he would definitely have approved….

## Statistical Analysis of the 1919 Eclipse Measurements

Posted in Bad Statistics, The Universe and Stuff with tags , , , , on May 27, 2019 by telescoper

So the centenary of the famous 1919 Eclipse measurements is only a couple of days away and to mark it I have a piece on RTÉ Brainstorm published today in advance of my public lecture on Wednesday.

I thought I’d complement the more popular piece by posting a very short summary of how the measurements were analyzed for those who want a bit more technical detail.

The idea is simple. Take a photograph during a solar eclipse during which some stars are visible in the sky close enough to the Sun to be deflected by its gravity. Take a similar photograph of the same stars at night at some other time when the Sun is elsewhere. Compare the positions of the stars on the two photographs and the star positions should have shifted slightly on the eclipse plates compared to the comparison plate. This gravitational shift should be radially outwards from the centre of the Sun.

One can measure the coordinates of the stars in two directions: Right Ascension (x) and Declination (y) and the corresponding (small) difference between the positions in each direction are Dx and Dy on the right hand side of the equations above.

In the absence of any other effects these deflections should be equal to the deflection in each component calculated using Einstein’s theory or Newtonian value. This is represented by the two terms Ex(x,y) and Ey(x,y) which give the calculated components of the deflection in both x and y directions scaled by a parameter α which is the object of interest – α should be precisely a factor two larger in Einstein’s theory than in the `Newtonian’ calculation.

The problem is that there are several other things that can cause differences between positions of stars on the photographic plate, especially if you remember that the eclipse photographs have to be taken out in the field rather than at an observatory.  First of all there might be an offset in the coordinates measured on the two plates: this is represented by the terms c and f in the equations above. Second there might be a slightly different magnification on the two photographs caused by different optical performance when the two plates were exposed. These would result in a uniform scaling in x and y which is distinguishable from the gravitational deflection because it is not radially outwards from the centre of the Sun. This scale factor is represented by the terms a and e. Third, and finally, the plates might be oriented slightly differently, mixing up x and y as represented by the cross-terms b and d.

Before one can determine a value for α from a set of measured deflections one must estimate and remove the other terms represented by the parameters a-f. There are seven unknowns (including α) so one needs at least seven measurements to get the necessary astrometric solution.

The approach Eddington wanted to use to solve this problem involved setting up simultaneous equations for these parameters and eliminating variables to yield values for α for each plate. Repeating this over many allows one to beat down the measurement errors by averaging and return a final overall value for α. The 1919 eclipse was particularly suitable for this experiment because (a) there were many bright stars positioned close to the Sun on the sky during totality and (b) the duration of totality was rather long – around 7 minutes – allowing many exposures to be taken.

This was indeed the approach he did use to analyze the data from the Sobral plates, but tor the plates taken at Principe during poor weather he didn’t have enough star positions to do this: he therefore used estimates of the scale parameters (a and e) taken entirely from the comparison plates. This is by no means ideal, though he didn’t really have any choice.

If you ask me a conceptually better approach would be the Bayesian one: set up priors on the seven parameters then marginalize over a-f  to leave a posterior distribution on α. This task is left as an exercise to the reader.