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Subject: Re: method of separating angles?

On Sat, 27 May 2006 16:13:31 GMT, "Dave Eberly"
>I have always wanted to pilot a jumbo jet. Would you
>be so kind as to tell me how to do this? I need a gentle
>explanation because my flying skills are poor. My only
>experience is driving really fast in my automobile!

The (apocryphal?) story goes that Euclid was teaching Ptolemy I
geometry, using the "Elements". The king (ruled Egypt 305--285 BCE)
complained about the difficulty and asked if there was an easier way.
Long generations of students know the answer given:

"There is no royal road to geometry."

Well, you can't blame someone for asking, eh? And Ptolemy's true
respect for learning is shown in his founding of the great library at

But now, as then, there is no use in students complaining. Sometimes a
gifted teacher can offer insights that for some students may seem like
a royal road. Enjoy. Yet most learning and most insight is generated
by the hard work of the student, for which there is no substitute.

All teachers are former students, and understand all too well what it
is like to struggle with a topic. We are also students of teaching,
and get pretty good at recognizing the difference between an honest
struggle and slacking off.

Dave highlights what we all know from practical experience about the
slackers: they suck valuable time and resources away from those who
would benefit more, and pose an ongoing hazard to those around them if
placed in positions of responsibility.

We'll go to extra trouble to help the strugglers, but cut the slackers
as quick as we can.


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