Eureka!  I see it.
...or do I?
Some personal reminiscences...

Freshman year in college

The year was 1949.  I was a new freshman at the school reputed to be the toughest in the state of Colorado.  Mines at Golden, conveniently also the home of Coors beer and Coors (scientific) ceramics.   "The freshman year will cut about a third of the freshman class," we were told.  Chemistry and math would do it.  "The sophomore year will cut about a third of those left."  Physics and math would do it.  "What fraction of those here now will be left at the end of your sophomore year?" we were asked.  We were told that if we couldn't come up with the right answer in a few seconds, perhaps we should consider D.U. (University of Denver, reputed to be a party school.) right now and avoid the agonies of  chemistry, physics and math at Mines.


Chemistry was formidable.  A large marble plaque over the laboratory door, looking like is was perched ready to drop on some unlucky freshman, commemorated the deaths of two faculty members who were poisoned by hydrogen sulfide when they didn't use the fume hoods properly.  The plaque couldn't have weighed less than 500 pounds, and the building was an unreinforced masonry structure whose age was more in the domain of the Geology Department than the Structural Engineering Department.  And if the physical hazards weren't enough to send a freshman running off to D.U.,  the course work would.  Scuttlebutt had it that the only way to get passing grades on the chem lab exercises was to take your unknown samples to a lab in Denver and pay  a professional to do the analyses.  For me, the classroom lectures were pure mystery.  I could only rarely see what the prof was driving at.

Fortunately, I met some fellow freshmen who seemed pretty bright and wanted to gather, especially before exams, and help each other figure things out.  It was a "study group," like in "Paper Chase."  We did something that was to be the key to a pretty successful undergraduate education.  We watched our fellow freshmen taking their chem lab samples off to Denver, sneak into the surveying course area after dark to measure the pacing exercise with a surveyor's chain, and rote memorize rituals and "formulas" to get answers to exam type problems which the profs had intended to be illustrations of concepts that could solve problems without a lot of rote memorization.

We made a great effort to master those concepts.

Mastery was different, but easier, than what we soon began to call "that engineering mentality."  All that learning without understanding and those acts of desperation a majority of freshmen used to get through the courses, generally aiming only at that diploma.  Trips to Denver, copying home work papers of those who could solve the homework problems, elaborate creative cheating on exams, and, of course, endless rote memorization without understanding.  (Endless trips to the free beer bar at Coors, too.)   If you asked an upper classman about a freshman course, you would get only one answer: "I've got credit in that course."

We had discovered that once you "saw" some principle, you wondered how you ever went without seeing something so obvious.  That rote memorization seemed to be ridiculously difficult when such simple "understandings" were there to discover.

We had discovered "Eurekas."

I eventually realized that chemistry seemed so difficult because I was expecting sophistication beyond the chemistry I took in high school, and at a level that simply wasn't there in our freshman course.  (My chemistry professor pointed this out to me after a couple of exams on which I had tried to give answers to the depth I was expecting.  He assured me that the high school chemistry was perfectly good in his course.  He also said I should be doing a lot better on his exams.)

Our study group got a reputation for "deriving every formula" on exams and were considered by most of our classmates as a bit daft for doing everything the hard way.  Our home work was also in great demand for copying.  ("Cooperate and graduate.")

Sophomore year: physics

Before every exam, our study group met in an empty classroom for at least an hour or two, and we quizzed each other in depth until we felt we had enough Eurekas under our belts.  Mechanics was easy.  Thermodynamics and electromagnetism were always a little obscure.  (I think Ken, the one of our group who went on to teach geology at Princeton and roam Nevada with author John McPhee, got a few more Eurekas on thermo than the rest of us.  Roger, the one who went on to work on nuclear weapons design, didn't even need Eurekas on math: he just "saw" that undergraduate level math.)  But there remained some very simple basic concepts that just didn't seem very clear.

One of these was "energy."  I had a gut level sense that our textbook definition of energy, "Energy is the capacity to do work," wasn't satisfactory.  I couldn't use it for anything.  I couldn't see just what energy really is.

I went to the school library to check the several shelf-feet of elementary, and not so elementary, physics texts there.  I looked at them all.  About half gave that "Energy is the capacity to do work" definition.  The other half seemed to be sidestepping the issue.  They gave good definitions of everything else I looked up—momentum, temperature, pressure, force, acceleration, viscosity, conductance, permitivity: you name it; they had it—but they seem to be consciously avoiding a definition of "energy."

Sneaky!  But why?

I gave up and forgot about it.  I would just have to give a rote memorized answer if a prof asked for a definition of energy.  But none did.  Just do the physics as needed and let the philosophers worry about it, was my sense then.  It did, nevertheless, always remain in the back recesses of my mind, giving me a sense of incompleteness.

Years later: I meet a curious character

By 1956, I was working in a semiconductor research laboratory in Southern California.  Every Wednesday, a physicist from a local college came around and gave a seminar on any topic we asked for.  He had an astounding gift for explanation.  He also had a magical gift for seeing "Eurekas" and solving really difficult problems.

I recall that among the many wonderful insights he sprung on us in answer to our questions, was the answer to my puzzlement over energy in that Colorado School of Mines library.  I remember a distinct "Aha!"  But I also recall that, even though it was crystal clear at the time,  I wasn't so sure about it several months later when I tried to recall it.  It wasn't something you use every day, and only much later did I find I needed it.  It wasn't quite there.

One of the eeriest of these slippery Eurekas, was something he showed us when he gave a complete course in differential equations (in one session).  He spent a few minutes covering the standard methods taught in a standard course.  Then he showed us his method:  guess an answer, plug it in, then adjust it to make it work better.

It worked for him.

He asked us for equations to solve.  We gave him the worst we could think of.

Things like, if D = the differential operator, d/dx, what y(x) satisfies:

He would solve them!  (And in an ambiguous equation like this one, he would solve all!)

Nobody else could, however.

The "slippery Eureka"?  While he was covering differential equations (in one session), he incidentally showed us a very simple, but elegant, way to differentiate any function, no matter how mathematically pathological.  As he described it I wondered how I could ever have done any differentiating without seeing his method.  It was utterly obvious, utterly trivial, utterly magnificent.

That was over 40 years ago.  Thirty years ago I had need to use that method.  I couldn't remember it.  I still can't.  Eerie!

The one incident that really impressed me more than any other happened when I went home at noon for lunch and the new Scientific American was in my mail box.  On the cover was Edwin Land's experiment in color photography.  Land had obtained full color images from only two separate black-and-white images.

I took the issue to the lab when I returned after lunch and showed it to my colleagues.  Everyone agreed.  This was puzzling and seemed to invalidate the three-cone theory about human color vision.  It was Wednesday.  The day of the seminar.  So we took the copy of Scientific American over to the main plant where our theoretical physicist met with us and showed it to him when he walked in the door.

He glanced over it.  He listened to our concerns about three-factor color, and thought for perhaps a whole minute: long for him to come up with an answer to a problem like this.  Then he surprised us with his opinion, "I don't think the three-factor theory of color is in any danger."

We asked him why he thought so.  He said he would have to think more about it, but he just had a feeling he was right.

We had a feeling he was wrong.  Even though he wasn't wrong that often, except perhaps about really esoteric physics.

About a year later an issue of IBM's research journal came out with an article on Land's color work.  An extensive computer analysis had shown that it was consistent with three-factor color theory.

I was very impressed with our seminar leader's insight.
Fortunately, a few years after I left Hughes (Aircraft Co.) Semiconductors our seminar leader (from Cal Tech) published many of his remarkable insights—his unique Eurekas—in a series of three physics textbooks, which are right now on the bookshelf next to me, ragged, worn, and dog-eared.  (Those big red books became the most popular books at the Cal Tech book store.)  Some of his color perception analysis, as it relates to dimensionality (in this case, to colorblindness), is elsewhere on this Web site.  So is his answer to "What is energy?"

Yes, our seminar leader was Richard Feynman.

The Feynman Lectures on Physics

  Dimensionality of color perception
What is energy?

When you are teaching: Don't declare; demonstrate.
When you are learning:??

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