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.