390 Quick Answers 5 February 

I have many done.  I will probably get project topics processed before I go home today.  Some may need to find a new topic for one reason or another (taken, or not meeting the intent).

Ok, today's our fifth day, it's not too early to ask for potential exam topics …

Overall context in all we do, from a student question in years past:  “would most mathematicians have been related to someone in the government or royal family, because I’d imagine it takes a lot of expensive schooling.”  Definitely “elites” in some way.  We will see exceptions, but this is worth admitting throughout our exploration.

One more overall context - when I look at original sources, this is the original writing of the historical mathematicians.  We have looked at something of this every day, I believe.  We will do this today with the Nine Chapters.  Someone asked "how did Archimedes do this?" the answer is - like we saw.  


Lecture Reactions

It's worth pointing out that short of using calculus, you don't really _have_ another proof of the area formula for the circle. 

It is easier to change a day every once in a while than to change a month.  Changing a month follows the moon more closely.  You probably agree that we've lost the connection between the moon and months.  When Intercalaris was used, the year was 355 days without it.  It intended to be used every 2 or 3 years, but it was neglected.  It ended with the year of (last) confusion when the Julian calendar began in 45 BCE.  Once it was pointed out, the Romans learned how to count leap years correctly. 

Spherical geometry is a non-Euclidean geometry.  It is worth remembering that, but it is also more different from Euclidean than hyperbolic is (for those who know, for those who don't - we'll get there).  Spherical geometry is more motivated by studying the earth and astronomy (celestial sphere), than studying the logic of alternatives with Euclidean.

The problem with Hypatia and the earlier Pandrosion is that we lack information.  Here's a different way to view both Hypatia and Pandrosion … neither of them made progress for women in mathematics.  Nothing changed because of them.  I'm willing to wait and give significant credit where we know it is due.  I'm not convinced either of these ancient examples are a big deal, nor that there aren't others doing more at the same time.  We just don't have evidence.  Good question - is there anything noteworthy about what either Hypatia or Pandrosion did that is not connected to them being women?  I would say … not much at best.  Why do we mention them?  Because they are famous, well at least Hypatia is.  We can't just ignore something that is widely talked about.  We need to at least address it.  Hypatia existed, was a woman, and did write and comment on mathematics.  That's true and known.  That's about what we know for sure.



Reading Reactions

I see a parallel between Chinese history and ours.  We change leaders regularly and lament problems without seeing that perhaps the underlying system is the problem.  It seems now to be growing common for new US presidents to spend their early time undoing all their predecessors did.  Emperor Shih Huang Ti took that to an extreme.  Maybe they would do that today if they had the power.  I'm sure some would.

Of course we don't write Chinese names in Chinese (the same goes for much that we will study), so to do so we need a way to transliterate, not translate, but to write using our letters.  There are two systems.  Suzuki uses the older, which would write "Hsia".  This is Wade-Giles.  The newer is Pinyin, which would write "Xia".  Neither is right or wrong.  Actually, I notice that Jeff is inconsistent about this, as he does use "Qin" and not "Ch'in". 

This is our first view of a base ten numeration system.  Alternating places helps with a lack of zero, but not completely.  Jeff obliquely hints that we don’t have evidence that our system can be traced to China - this is true and worth clarifying.  China develops some of the main ideas, but we don't get our system from them.  We'll talk more about that next week.  It is a big step and a big deal.  Much of their algebra is by successive approximation.  Because of this, it mattered more if an answer was good enough than if it was correct.  Therefore they had little concern about termination or not.  If you only work with decimals and not fractions, there is no noticeable difference between irrational numbers and non-terminating rational numbers.  The Chinese were definitely working with something equivalent to decimals as you know them.

There's a small mention of zero in this section.  I will say more about it next time with India and some recent (2017) developments in history that are clearly not in the book.  

Successive edits and additions to the nine chapters throughout the centuries is definitely building on past work.  It is _as important_ as Euclid's elements, *NOT* based on Euclid, which they surely did not have access to.

Chinese mathematics states results, and sometimes methods, but typically not reasons.  It just wasn't their choice of focus. 

Yes, the Chinese were working with modular arithmetic in the Chinese Remainder Theorem.  No, they didn't use the symbols we do now.  Notation is almost never what is most important.  Learn to see past it. 

Chinese work with negative is impressive at this time, and it is curious their colour choices are switched from our current choices. 

Yes, it seems Marco Polo's accounts are not credible.