Kepler, Galileo, Newton, Einstein: not a bad roll-call

The following is an edited extract from notes which I give to students before going through the derivation for the rather intimidating equation below.

You have just arrived at an equation which bookmarks a seminal moment in the history of science.


Around this time (16th century) an astronomer called Johannes Kepler discovered empirically (i.e. by analyzing data on the motion of planets) that the square of the periodic time of these planets (time for one complete orbit around the sun) is proportional to the cube of their distance from the sun.
Kepler actually stole the necessary data from a colleague, Tycho Brahe, but that’s nothing new in the world of Science. We will conveniently ignore that for now.

Later on Newton came along and was able to demonstrate this relationship mathematically, by combining a well known equation for circular motion on Earth with his own universal law of gravitation. We are about to follow in his footsteps and see exactly what he did and how he did it. Do not under-estimate the importance of this exercise (yes you have to know it for exam purposes, but that’s not why I consider it important).

This event had two very important consequences.
1. It showed that Newton’s Law of Gravitation must be valid in its own right, which was very important in securing Newton’s reputation as a giant of science, both at the time and for posterity.
2. Even more importantly, it demonstrated that ‘the heavens’ followed the same rules of science as those which operated here on Earth.
This meant that they were a legitimate area of study, and so Astronomy (which in turn led to Cosmology) was given an added respectability. Just to give a sense of what people believed at the time, Kepler had to spend much of his time during this period defending his mother of charges of being a witch.

I can think of no modern discovery which compares with this. Even if we discovered life on Mars it really wouldn’t be that big a deal. For up to this point the heavens were considered off-limits – the realm of God or the gods or whatever you’re into yourself. But now they could be shown to be just another series of objects which followed set rules, much like cogs in a complicated clock. So God was being pushed into the wings. You could see why neither Martin Luther or the Vatican Church would have been keen fans.

Kepler was following on the work of Nicolas Copernicus (known to science students down the ages as ‘copper knickers’), who in turned showed that the Earth revolved around the Sun, not the other way around.
Galileo’s run in with the Church was because he supported Copernicus’ view, so Galileo never actually made that discovery but was happy to use it to make fun of the church authority figures of the time. I think we all know how that worked out for him.

This was really the dawn of science, and progress was hindered by medieval views of the astronomers themselves. It took Kepler many years to realise that the orbit of the planets was elliptical in nature, not circular. He had assumed initially that the motion had to be circular because a circle was a perfect shape (harping back to the teachings  of Pythagoras and Aristotle, among others) and therefore would have been more pleasing to God who obviously had created the planets in the first place.

Similarly Newton, despite being heralded as one of our greatest ever scientists, spend up to 90% of his time trying to date the creation of Earth by tracing who gave birth to who in the bible.

But then Newton had another problem. He realised that Kepler was correct in stating that the planets traced out elliptical orbits, but even Newton’s equations didn’t fully match the path of the heavenly bodies; according to Newton’s equations the planets should slowly but exonerably drift from their current pathways. He couldn’t figure out why this didn’t happen – after all, his equations seemed to be perfect in every other way. And Newton believed that he was getting his ideas directly from God. Which doesn’t leave much room for admitting you made a mistake.

We now know that while Newton’s equations are very accurate, we actually need Einstein’s Theory of General Relativity to explain why they don’t precisely describe the motion of the planets.
It’s interesting to note that Newton’s explanation was that God must step in every so often to gently nudge the planets back into their preferred orbits. Now as you now know, invoking a deity to explain discrepancies in scientific observations is the antithesis of Science. So perhaps Newton wasn’t actually so mighty after all. This is partly why he is sometimes referred to as the last sorcerer rather than the first scientist.

So now we’re up to Einstein. His general theory of relativity suggested that the universe was expanding, but just like all of his predecessors he was a man of his time, and this coloured how he saw the world. It was believed at the time that the universe had always been the way it is now (this is referred to as the ‘Steady State’ theory). Einstein figured that there must be some mistake in his paper so he introduced what he called a ‘cosmological constant’ which basically amounted to a fudge factor which altered the implications of his calculations and prevented the universe from expanding.

Which was all very well until Hubble (he of the ‘Hubble’ telescope) showed that the universe was actually expanding after all.

Einstein referred to this as his greatest ever blunder.

So there you have it. This has been my attempt to put some context on the derivation that we are about to carry out. It is our chance to repeat one of the greatest moments in the history of science.

So you have two options; you can consider this exercise to be a pain in the ass or you consider it an incredible privilege to be in a position where you can follow in the footsteps of giants.

I think we know which option I go with.

And don’t be afraid to tell your parents this tonight; they may well throw their eyes up to heaven but if they do that’s a slight on them – not on you.


Why do we remove Wonder from Science Education?

If it’s possible to dedicate blogposts to individuals then I choose to dedicate this to my aunt; Sr Cathy. Like many religious folk I know, her passion for Science may well surpass her passion for her religion. Or maybe she’s just passionate about everything. Either way, I’m looking forward to meeting up with her over the Easter break as part of a big extended family celebration.

Wonder is a theme we return to again in again in this blog. More specifically the theme is one of frustration that we have deliberately removed all reference in our science textbooks and syllabi to concepts that evoke a sense of wonder. And it doesn’t help that it seems to bother so few other people. Which is why every time I come across somebody else expressing the same frustration I move to wrap the up in cotton wool and store in away in s0 that I can return to it anytime I need reassurance that it’s not just me. And where better to store it than here?

Students today are often immersed in an environment where what they learn is subjects that have truth and beauty embedded in them but the way they’re taught is compartmentalised and it’s drawn down to the point where the truth and beauty are not always evident.
It’s almost like that old recipe for chicken soup where you boil the chicken until the flavour is just . . . gone.

The speaker, David Bolinsky, is famous for having created an incredible animation on the private life of cells. I have watched that video many, many times (it’s a beauty in it’s own right) but it was only when I watched its Bolinsky talk about it on TED that I zoned in on his quote above.

I devour popular science, finding its history and its wonder a constant delight. . . . It is a mystery how so many science teachers can be so bad at their jobs that most children of my acquaintance cannot wait to get shot of the subject. I am tempted to conclude that maths and science teachers want only clones of themselves, like monks in a Roman Catholic seminary.

That was from Simon Jenkins in the Guardian

We are deprived by our stupid schooling system of most of the wonders of the world, of the skills and knowledge required to navigate it, above all of the ability to understand each other. Our narrow, antiquated education is forcing us apart like the characters in a Francis Bacon painting, each locked in our boxes, unable to communicate.

That was courtesy of well known columnist George Monbiot

The way I was taught science made it feel like nothing more than a series of disconnected facts – the eureka moments of long dead scientists. My knowledge of Einstein’s work by the time I went to university was E=mc2; something like the Einstein-Silárd letter was completely absent from my education. I learned more about the history of nuclear physics from the play Copenhagen than I ever did from a school discussion.

Andrew Holding

We educators take this incredibly exotic jungle of knowledge called Science and distil it again and again until all the wonder has been removed! We are left with nothing but a heap of dry shavings. We then pour this drivel into our syllabus and textbooks and make our students learn it off by heart so that it can all get vomited back up come exam time.
And then we wonder why so many young people don’t like science.

That one’s mine.

It’s really such a shame that the wonder of Science only seems to be spoken about by artists, poets and writers. Why do scientists (and science teachers, and in particular those who are responsible for drafting the science syllabi) hide from it so much?

Anyway, the reason for this particular post is that it’s time to add the opinion of the author of what is for me the greatest book ever written in the Popular Science genre; Bill Bryson, author of A Short History of Nearly Everything.
I’ll paste in the short quote first, but to understand the context it deserves to be read in its entirety so I’ll follow with that (and anyway, reading Bryson could hardly be termed a chore).

It was as if he [a science textbook author] wanted to keep the good stuff secret by making all of it soberly unfathomable. As the years passed, I began to suspect that this was not altogether a private impulse. There seemed to be a mystifying universal conspiracy among textbook authors to make certain the material they dealt with never strayed too near the realm of the mildly interesting and was always at least a long-distance phone call from the frankly interesting.

Here is the full context:

My own starting point, for what it is worth, was a school science book that I had when I was in fourth or fifth grade. The book was a standard-issue 1950s schoolbook – battered, unloved, grimly hefty – but near the front it had an illustration that just captivated me: a cutaway diagram showing the Earth’s interior as it would look if you cut into the planet with a large knife and carefully withdrew a wedge representing about a quarter of its bulk.

It’s hard to believe that there was ever a time when I had not seen such an illustration before, but evidently I had not for I clearly remember being transfixed. I suspect, in  honesty, my initial interest was based on a private image of streams of unsuspecting eastbound motorists in the American plains states plunging over the edge of a sudden four-thousand-mile-high cliff running between Central America and the North Pole, but gradually my attention did turn in a more scholarly manner to the scientific import of the drawing and the realization that the Earth consisted of discrete layers, ending in the centre with a glowing sphere of iron and nickel, which was as hot as the surface of the Sun, according to the caption, and I remember thinking with real wonder: ‘How do they know that?’
I didn’t doubt the correctness of the information for an instant – I still tend to trust the pronouncements of scientists in the way I trust those of surgeons, plumbers, and other possessors of arcane and ¬ privileged information – but I couldn’t for the life of me conceive how any human mind could work out what spaces thousands of miles below us, that no eye had ever seen and no X-ray could penetrate, could look like and be made of. To me that was just a ¬ miracle. That has been my position with science ever since.

Excited, I took the book home that night and opened it before ¬ dinner – an action that I expect prompted my mother to feel my forehead and ask if I was all right – and, starting with the first page, I read.

And here’s the thing. It wasn’t exciting at all. It wasn’t actually altogether comprehensible. Above all, it didn’t answer any of the questions that the illustration stirred up in a normal enquiring mind: How did we end up with a Sun in the ¬ middle of our planet and how do they know how hot it is? And if it is burning away down there, why isn’t the ground under our feet hot to the touch? And why isn’t the rest of the interior melting – or is it? And when the core at last burns itself out, will some of the Earth slump into the void, leaving a giant sinkhole on the surface? And how do you know this? How did you figure it out?
But the author was strangely silent on such details – indeed, silent on everything but anticlines, synclines, axial faults and the like. It was as if he wanted to keep the good stuff secret by making all of it soberly unfathomable. As the years passed, I began to suspect that this was not altogether a private impulse. There seemed to be a mystifying – universal conspiracy among textbook authors to make certain the material they dealt with never strayed too near the realm of the mildly interesting and was always at least a long-distance phone call from the frankly interesting.

I now know that there is a happy abundance of science writers who pen the most lucid and thrilling prose – Timothy Ferris, Richard Fortey and Tim Flannery are three that jump out from a single station of the alphabet (and that’s not even to mention the late but godlike Richard Feynman) – but, sadly, none of them wrote any textbook I ever used. All mine were written by men (it was always men) who held the interesting notion that everything became clear when expressed as a formula and the amusingly deluded belief that the  children of America would appreciate having chapters end with a  section of questions they could mull over in their own time. So I grew up convinced that science was supremely dull, but suspecting that it needn’t be, and not really thinking about it at all if I could help it. This, too, became my position for a long time.


Everything a Primary School teacher (or student) needs to know about gravity. And then some.

This post is in the context of a question posed by a primary teacher on a forum recently. Rather than reply there I thought it safer to do so where I could offer a more comprehensive answer.

We tend to associate the concept of gravity with the English scientist Isaac Newton who lived in the seventeenth century.
But he didn’t ‘invent’ gravity; objects were falling to earth long before Newton arrived on the scene, so what exactly did he do?

He did what so many other kids do; he asked asked a silly question. ‘Why do things fall down?’
It does seem like a silly question, which is why nobody took it seriously before, but when you think about it it’s actually quite profound; how does the apple in an appletree ‘know’ which way to fall? How does the earth ‘know’ (if it pulls the apple down) that the apple is there in the first place ? Newton was never able to answer that question. He famously said  “Hypotheses non fingo” (Latin for “I feign [frame] no hypotheses,” or in other words, “I haven’t a clue why this works the way it does”). It’s not like there’s a string connecting the two objects, but yet the apple acts as though there were indeed an invisible string pulling it downwards.
What form does that invisible string take?

I don’t know the answer, but I do know that scientists haven’t fully worked it out yet either.
It has been suggested that all objects exchange particles called ‘gravitons’ and it is as part of this exchange process that the objects come together. The problem is that these gravitons have never been detected.

Another possibility is gravitational waves. These were postulated by Einstein in his Theory of General Relativity. There has been some indirect evidence for these but again they haven’t yet been detected directly. We know we don’t know all there is to know about gravity, and to suggest otherwise would be to do a disservice to your students. In fact the same holds for a lot of science. Gravity does seem to be a little like magnetism, yet the rules which govern gravity don’t work for magnetism and vice versa. The holy grail of physics is to show how the rules that govern the motion of very large objects like planets is connected to the rules that govern the operation of very small objects like atoms. And there’s absolutely no reason why one of your students can’t be the one to make this connection and win their very own Nobel Prize (with a bit of luck they will acknowledge  you  in their acceptance speech as the spark which ignited their passion for Science).
Matthew is a former student of mine and is currently doing a PhD with NASA on this topic. I asked him to explain it to me:

“In the Einsteinian framework, however, gravity is not a force but a curve in space-time. So any object with mass induces a curve in the spacetime around it. Any other object no longer travels along a flat spacetime, but along a curved path. That’s essentially what’s happening to the apple. Instead of hovering at the end of the branch as it would in a flat spacetime, the ‘forward direction’ of spacetime is curved due to the Earth, so the apple just follows that curve, which in three spatial dimensions is just a straight line down.”

Watch the following clip for a wonderful demonstration of a curving space-time –  imagine doing this with your kids in class: you can tell them you are studying Einstein and doing Rocket Science.

But while Newton couldn’t say why gravity worked, he was able to quantify the force of gravity, i.e. he was able to devise a formula which now enables us to say how big the force of attraction will be between any two objects. It depends on how big the objects are (or more specifically their masses) and the distance between them.

It turns out that any two objects will exert a gravitational pull on each other. Now this is mad. It means that there is a force of attraction between you and your biro, and if it was just the two of you floating in space with no other objects or planets in existence, that force of attraction would result in the biro moving towards you and you moving towards the biro. Similarly there is a force of attraction beween each student and the student next to them (cue lots of giggles) and the bigger the size (or mass) of either student, the bigger will be the force.

Newton also established that the force that kept the planets in orbit around the sun was the same force as that which pulled the apple to earth. This idea was a big, big deal at the time. It meant that the planets followed the same laws of physics as objects on earth. Prior to this ‘the heavens’ were thought to be the realm of the gods or God and therefore not subject to our analysis but after Newton they were seen as fair game for anybody to study. I don’t think there’s any way we can really appreciate how big a deal this was. And while Newton wasn’t the very first to realise this, he was the first to demonstrate it mathematically.

The following is a nice video which outlines the significance of Newton and Einstein to our understanding of gravity. You only need the first ten minutes.

The bottom line for me is that you have an incredible audience who will lap this stuff up. Please, please don’t play down the mystery or the wonder. That, unfortunately, is what happens at second level and I have been trying to get teachers to fight it my entire professional career, with very limited success (it doesn’t seem to bother many other teachers, but I have it bad).

Don’t allow your lack of technical knowledge to put you off engaging with the material. Remember when it comes to Science nobody, and I mean nobody, has all the answers. If we’re looking to turn some of these kids into scientists then what they need more than anything else is curiosity and a good old-fashioned sense of wonder. If you can help develop that then everything else will follow.

My contribution to Science Week – I thought I might teach some physics

At 40 mins long it’s not going to go viral anytime soon. It’s the middle 40 minutes of a double class but in it we managed to learn about some of the following:

The structure of the atom.

We, and everything around us, are mostly empty space.

We discovered that the appearance of  ‘solidness’ is an illusion – which lead to a  discussion about how light works.
We learned that there is a cultural aspect to what we see (and you definitely won’t find that in physics textbooks) and that Newton himself was subject to this and it resulted in him making a boo-boo that still goes uncorrected right up to today.

We discovered that electrons are constantly cascading down along everything we see in a seemingly never-ending avalanche, powered by energy from incoming light (so when this power source disappears, the electrons no longer have energy to jump up or fall back down, otherwise known as darkness).

We learned why things feel solid – all to do with the force of repulsion between electrons at the surface.

We developed a deeper understanding of Newton’s Third Law.

We discussed the fallacy of language – know the word for something (like gravity) and understanding what gravity actually is are two very different things, and shouldn’t be confused with each other.

We discovered that physics teachers don’t have all the answers, and should never pretend otherwise.

We were reminded that because almost none of the above is in the syllabus, the syllabus is a disgrace. It’s no wonder students don’t see the point of it.
There were 22 students in that class and the discussion could have gone on and on – I had to kick them out the door.  One can only imagine the conversations they must have had over the dinner table that evening.

If only all those who make such a fuss over Science Week could put a fraction of that effort into making the school syllabus a source of wonder and curiosity instead of what it is – a series of dull as dishwater facts which are to be merely learned off by heart.

Why does Science Week bug me so much?

What is it about Science Week that gets under my skin so much?

It seems to be the one week in the year where we are supposed to go out of our way to make science interesting; the corollary being that for the rest of the year we concentrate on ‘normal science’ which isn’t interesting. We go back to ‘the study’ and our drive for ‘good results’.

This idea is reinforced when we look at the various syllabii that are in play. There is nothing in the preamble or the main section of any of these about emphasing the wonder in the subject or indeed even encouraging a sense of curiousity – which is what Science Week is all about.
I have written about this before and penned the following few lines to sum up my frustration.

We educators take this incredibly exotic jungle of knowledge called Science and distil it until all the wonder has been removed and we are left with nothing but a heap of dry shavings. We then pour this drivel into our syllabus and textbooks and make our students learn it off by heart so that it can all get vomited back up come exam time.
And then we wonder why so many young people don’t like science.

How about if, when drawing up a new syllabus, we use WONDER as our central idea? It would probably mean that when teaching biology we would actually have to discuss evolution (the word doesn’t exist on the current junior cert syllabus – can you believe that?). It would mean having to teach about topics in cosmology – this currently doesn’t feature at either junior cert or leaving cert physics level, despite it being one of the main sources of interest to students of all ages, and also a prominent feature of every Science Week.
In fact in just about every topic at both JC and LC level the content could and should be build around instilling a sense of awe rather than consisting of a series of dry facts.
I am currently teaching The Electron to leaving cert physics students. In an earlier topic we proved that light was a wave by demonstrating interference of light waves. In this topic we prove that it is particle-like in nature by demonstrating the photoelectric effect. Both of these demonstrations need to be known for exam purposes and presumably most ‘good’ students learn them without thinking much about them. To read about these in either the textbooks or the syllabus you’d think that there was nothing of particular interest here when in fact these two contradictory phenomena are cornerstones in possibly the greatest movement in physics of all time: what is now known as quantum physics. Quite simply, you can’t have something which is both a particle (being in one specific place) and also a wave (being spread out) – yet that’s exactly what we find light to be. To quote Einstein “The more successful quantum physics gets, the sillier it looks”. But then if you’re reading this far you probably already know quite a bit about quantum physics and how utterly wonderful it all is. So you’ll know why I am baffled as to why all the fun has been ignored.

We could do the same for almost any topic on either the junior cert or the leaving cert course. But then that would be a bit radical. Best to leave all the boring stuff in and leave the fun stuff for Science Week. The word ‘wonder’ has most likely never featured in any science syllabus over the past four hundred years, any where in the world, so why change now?

What also bugs me is why so few other teachers seem to care about this. I know many of them introduce the wonder associated with the concepts as they teach it, but many others unfortunately don’t. And if we look at the number of students who drop Physics and Chemistry at the first opportunity it may be that the latter category of teacher represents the majority. What’s particularly puzzling is that if you go to any teacher conference they will usually have these ‘interesting lectures’ as part and parcel of the day, and no surprise for guessing that these are the best attended. So why don’t these same teachers make more noise about including interesting material on the formal syllabus? How can a biology teacher stand over a junior cert biology syllabus that doesn’t include the word ‘evolution’?

This is just the latest of my rants about the lack of wonder in Science education – for more see There’s that word again . . . WONDER

For a gentle introduction to wave/particle duality see the following:

Harry Chapin – Flowers are red

Criticising our education system is not new – why would it be when it’s like shooting fish in a barrel? One of the better known recent commentaries came from Sir Ken Robinson at a TED conference a few years back who made a very convincing argument for changing our focus away from the  academic subjects and instead develop a greater emphasis on the arts as part of our students’ formal education.

Sometimes the best critiques come not from ‘experts’ but from those well outside the academic circle. Harry Chapin’s Flowers are red  always been one of my favourite songs in this regard. It really doesn’t require anything more to be said. Listen for yourselves and if you’re a science teacher ask yourself which teacher you want to be like.
And then try to answer honestly which of the  two teachers your students would match you with.

Remember almost every student comes into secondary school with a deep sense of wonder which is all you should need to succeed in Science. Few leave with this passion still in good working order. We must at least allow for the possibility that we teachers are part of the problem.

Aims and Objectives won’t get us out of this one.

An Engineer’s Guide to Cats

For many years now many of my brightest and best students have gone on to study Engineering, despite my best efforts to the contrary.
I consider an engineer to be a physicist who has lost his sense of wonder.

By the way, do you know where the term ‘civil engineer’ comes from?
Apparently it was to distinguish engineers who worked outside the military from those who worked inside it.
Thanks to The Guardian for the heads up on the video.

The BBC – supporting Science

World of Wonder -Science on the BBC
This series is part of World of Wonder – a year of Science across the BBC in 2010. From popular science to Chaos Theory, there’s a little something for everyone.

Did you know that the BBC has a Science homepage?
It’s called “World of Wonder” – what a cool title.

You see the BBC know that if they want the public to watch their programs then they have to first of all draw them in and then they must deliver.
And the single best way to do this is with wonder. And the single best source of wonder is science.
And of course it helps if you have a passionate presenter who is comfortable in front of the camera.

The following represents just some of what they have produced lately:

The Story of Science: Power, Proof and Passion

Hammond’s Invisible Worlds

Museum of Life 

Journeys from the centre of the earth

The Power of the Planet

How the earth made us

South Pacific

Wonders of the Solar System 
Professor Brian Cox (the “rock star physicist” who looks as awed as I do when watching it) visits some of the most stunning locations on earth to describe how the laws of nature have carved natural wonders across the solar system.
Professor Cox has just started work on a new series entitled “Wonders of the Universe” – follow him on twitter.

Not a bad roll of honour – each one a wonderful representation of the story of science.

When RTE does Science it is invariably a natural history program – nothing wrong with that in itself, but a little more from the physical sciences would be nice.

Presumably it’s rather expensive to produce a Science program. The last wildlife program from the Nature department of RTE was aired recently: Wild Journeys was a three part documentary with wonderful scenary but woeful narration. It was completely devoid of that one word which the viewer could respond to more than any other – wonder.  Check it out for yourself

Next time RTE is looking for material for a Science program they should just ask Mary Mulvihill for pointers and then give a lot of thought to choosing a presenter (and not just go with whoever happens to be passing on the corridor at the time). Why not have somebody who isn’t afraid to sound amazed or in awe of Nature? Imagine someone like Kathryn Thomas, Tommy Tiernan or Ray D’arcy narrating, or better still in front of the camera rather than just providing the voiceover.

Or my personal favourite – physicist/comedian  Dara O’ Briain.

A step too far for the fuddy duddies  in RTE methinks.

Wonder in Science (why do we hide it?)

Yes, yes, yes, yes!

Just read this online article from Simon Jenkins in the Guardian

I devour popular science, finding its history and its wonder a constant delight. . . . It is a mystery how so many science teachers can be so bad at their jobs that most children of my acquaintance cannot wait to get shot of the subject. I am tempted to conclude that maths and science teachers want only clones of themselves, like monks in a Roman Catholic seminary.

I couldn’t agree more. It is a sense of wonder in the world around me that has drawn me into science, and yet wonder is the one thing that is sorely lacking from all text-books and school syllabii. And we as teachers are doing absolutely nothing about it. We should be ashamed of ourselves.

Listen to all the big-wigs tell us why we need more students doing science – it’s the economy, stupid. Yet ask any kid why they are fascinated with science and the economy is not likely to come top of their list of reasons. It’s that word again – wonder. So why are we afraid to tackle it at school level? And why does nobody talk about it?

Heaven preserve us from engineers, university professors and politicians getting their grubby mitts on another science syllabus. Not unless they can first demonstrate a proven track recond on rating wonder as highly as a kid does. Not that we teachers have much to boast about in that regard either. It’s as though we try to hide our sense of wonder because somehow it doesn’t seem appropriate. Is it because we teachers like to give the impression that we have all the answers and therefore there should be nothing to fill us with wonder. I honestly don’t know. And I’m not even sure what I can do about it.

But I guess a good aul’ rant wouldn’t be a bad place to start. 

Feynman, in this regard as in so many others, remains an inspiration.