390 Quick Answers 26 February

Reminder:  Diversity Summit tomorrow - discuss Friday, Annotated bibliography due Monday. 

Although this class is smaller than prior, I am very impressed that all but 2 have reactions done by 10p.  Nice job!

Lecture Reactions

This being 2024 relates to Dionysius Exiguus computing Eastre backwards from his time a full 19x28 cycle before the dates that were known before him (up to what we now call 532).  He did not use different starting points for years, but he inspired Bede to use his work to define our starting point. 

Zero was the newest and most unfamiliar digit. 

Gerbert was only asking how many times can we subtract 8 from 96, and subtracting 10-2 instead, and then continuing to the next step. 

Age for Alcuins problems - probably teenagers, although now many would be well suited for 8-10, I would think. 

Alcuin didn't explain his methods much, but he was mostly computing.   No variables.  We've done many things that act like variables "the thing" does this so "add one to the thing" &c.  But not actually using variables until Jordanus.  These things don't change immediately for everyone.  We'll see slow transitions.  Point of clarification:  we've seen plenty of variables for points in geometry.  That is different than variables for quantities.  Variables for points is not algebra, just a way to refer to diagrams. 

Algebra is still heavily geometric for a while, although that is starting to fade.  It will fade more on Friday. 


Reading Reactions

Yes, I believe I mentioned disease as of the causes of the dark ages in Europe. 

Bonfils didn’t make my cut, not much there it seems, but he’s talking about astronomical events for the calendar, and astronomy _is_ different in different locations. 

I’m not talking about ben Gerson’s trigonometry in lecture, so I’ll say something here.  We’ve seen (in Indian trigonometry) the work to get to 1 1/2° before.  Ok, some need reminders:  30° you know (I hope - from half of an equilateral triangle and the MCRTT).  18° comes from a pentagon.  We can use half-angle formulas to find sines for angles half of each of those.  Do that to get 15°.  Then we can use sine difference formulas to get 3°.  Then half again to 1 1/2°.  That’s not surprising.  ben Gerson then uses sine addition to get up to 16 1/2°.  Then half-angle six times to get 1/4 + 1/128°.  Similarly, since 15° = 16 - 1°, he uses half-angle six times to get 1/4 - 1/64°.  These can all be done exactly.  Then interpolate down to 1/4°.

Someone declares the value of money, especially when it isn't directly tied to the value of the metal it's made of.  And, even if it is, people can lie about that.  Just saying "coins are worth more" gives a false sense of value that devalues them. 

Until 1974, in England (10^6)^2 was a billion and 10^9 was a thousand million only.  Using the old system (10^6)^3 would be a trillion.  In the US and now the world 10^3 is a thousand, (10^3)^2 is a million, (10^3)^3 is a billion, and (10^3)^4 is a trillion.  The old English way seems more logical in a way.  Old way:  (10^6)^n = n-illion, US way:  (10^3)^(n+1) = n-illion. 

This isn’t a history of science class, so we’re not talking about ancient views on physics &c.  We’ll have a little here to give you a flavour, and we’ll talk about it when it connects deeply.  That also being said, Jeff makes many choices, and I kinda follow along, making more refined choices (i.e. choosing which of his stories to elaborate on).  We are passing through all of human history of mathematics - and we surely miss a lot.  Here’s a great thing - reading history of mathematics is much easier than reading mathematics in general.  So, if there’s something you want to know about 1. you can read about it, and 2. if you want help, I can help you.  For any reason whatsoever, I would be happy for any related conversations from this class - it doesn’t happen as much as I would like.