Here are the directions for Friday's exam
(it's in Milne 104 - come early). Please remember my goal in this
exam is to give you an opportunity to show me that you have learned
something. I truly believe you have. "Greco-Roman" is
sufficient as the distincition is a little confusing.
Someone asked about the history of proof methods. Most are *very*
old. Aristotle and Euclid surely understood proofs and
conradiction, and contrapositive. The only one that seems to
leave is induction, which we saw with Levi ben Gerson. We may see
it again. That's about it for main proof methods.
§7.1 Quick Answers
The Gregorian calendar is, finally, the calendar we now use today (with
the occasional leap second thrown in). I think Viete's main
opposition to this calendar is merely for change sake. Changing a
calendar is always a challenging thing - and even though better ones
could be used, people have traditions tied up in the calendar.
Ask me sometime about more of this story. I think NS and OS
was used mostly at the time and isn't now. Now we tend to use
Julian dates for before 1582, Gregorian for after 1752 (when Britain
and hence US as a colony changed). Dates between those are
definitely confusing, and Russian dates are confusing until the
revolution in 1917. Modern historians must be careful as to which
calendar is being used. Our vernal equinox moves a little bit
(one to two days - this year it's 3/20), based on the fluctuations
coming from leap days, but it averages constant. The church still
uses a fixed date (the ecclesiasitcal sun) for determining Eastre.
The one most important natural parameter in the calendar is the
year/day ratio - which is practically irrational and precisely not
constant. All these struggles deal with that fact.
Due to time constraints and other things being more important, I won't
say much about Viete's 45-degree problem. (For this and anything
else I skip in class - ask me outside and I'll be happy to talk to you
about it.) DeMoivre's theorem is (cos x + i sin x)^n = cos nx + i
sin nx. Viete was working with sin 45x, so it was related.
The problem was very carefully crafted. Viete needed extra
terms in his equations to make sure they were all the same dimension.
If he had a quadratic equation, everything needed to be two
dimensional. Some of his extra terms amounted to something like
putting units^2 on the end.
I don't think Fermat was secretive. Remember he was basically on
the French supreme court. He did mathematics in his spare time,
but it wasn't his career. I probably do agree with Suzuki that
Fermat is overrated, but then again I think everyone that everyone
thinks is great is overrated. Fermat is computing area under a
curve, but definitely doesn't think of them as integrals. He's
surely not the first to find areas under a curve.
Roberval worked with both areas and rates of change in his Treatise of
Indivisibles. I have some sources for this, but haven't looked at
it closely. He kept his secrets so that he could win challenges
to keep his job - not because he didn't want others to know.
NSA is the employer of the greatest number of mathematicians in the
world. Yes, coding is still a major pursuit of mathematics.
Coding also goes back to ancient times.
This is a cycloid.
"In Cardano's writing of 5x^2 +4x+ 8, he has a cos after the 4, what
does that represent?" "Cosa" is Italian for "thing". It was an early variable.
A conjecture is a guess that remains unproven.
Pascal's adding machine. This could only add and subtract.
Richelieu is perhaps best known as the antagonist in the Three
Musketeers. Please remember one of Suzuki's main goals is to
connect to things you already know. Another is to put some facts
around some popular myths.
It's very interesting that we seem to be at a dramtic point of change
in mathematics. We could talk about this endlessly, and I
wish we had time to do so, but I also value getting to go where we
will. Definitely ask me about these things sometime - I would
completely enjoy talking with any of you about it.