390 Quick
Answers 29 March
Drafts
of papers due in one week. The sooner you get to me the
sooner you will get back. To be more explicit about GREAT
Day rehearsals. You need to schedule by Wednesday 3
April. You need one the next week, and one the week
after. You may do them _before_ this, not after. If
you do not do two rehearsals on this schedule, you will not earn
credit for your presentation.
Today’s
a little different day - I will get ahead and preview some if 8.2
today. I will try to not spoil the stories.
Ok,
I want to say something about the sources I display in
class. I’m really sorry that they have so many typos.
This is definitely not a misunderstanding from the original
mathematician, nor is a problem with the translator. The
problem is one step beyond that. It is the typesetter of the
compilation that I pulled from. I know they’re annoying, and
they annoy me, but please try to look past them. (I promise
that I am not displeased with you for caring about them.)
Lecture
Reactions
Multiplication
is hard. I want to multiply 8.7654321 and 2.3456789 so
instead I take log(8.7654321) and log(2.3456789) using tables. Then I add the results
(adding it easy). That gives log(8.7654321 x 2.3456789).
So I use a table backwards to see what number this is
the logarithm of and that tells me the product is.
There is a (unusually large) slide rule above the math
dept. office. If you want to know how to use one or what it
has to do with logarithms, come ask me.
I
think there was a push at this time for a way to make
multiplication easier (and better than prosthaphaeresis).
This is likely why both Napier and Burgi found it at the same
time. This phenomenon of similar work happening at the
same time is common.
One
reasonable answer to the Newton-Leibniz debate is
“neither”. Calculus was growing slowly over
centuries. We’ve seen ideas long ago, and there are
several who took very serious steps. Newton and Leibniz
were the first two to present a systematic theory. On the
other hand both still struggled with some of the details.
In
Barrow’s FTC, remember the top function is an area accumulation
function. The top function measures how much area the
bottom function has accumulated up to that point. To be
more precise R times the top function gives the area. This
was done because a length is not an area, so R times the top
function gives the area. As far as I know none of these
people called it ‘the fundamental theorem’. Comparisons
between Leibniz and Barrow’s proofs: Leibniz used
infinitesimals and that the integral of dx is x. Barrow
avoiding the subject by working more geometrically with
tangents. Regarding my claim that R = 1, there was no
scale on the graph nor units. We may use any units we
choose. I choose to use units of R. We've seen this
before.
I don't know where Newton got his series for sine and cosine (it
would be interesting to know), but first guess would be same as
Taylor and MacLaurin. We'll see that today.
Reading
Reactions
I’m
*not* going to get into physics here, but I do want you to see
that gravity is a curious thing … how does the sun control the
earth when they are so far apart. The moon’s position became
important for use in navigation tables, as explained in the
subsequent paragraphs after saying it became important. The
moon is the simplest example of a three-body problem. We can
assume that the earth is only affected by the sun’s gravity, and
we do fine. But, to analyse the moon we cannot ignore either
the sun (because it’s so big) or the earth (because it’s so
close).
The
English are avoiding work in the calculus from fallout from the
Newton-Leibniz controversy. Nationalistic isolation does
harm. Curiously, we (here now) tend to learn more
Leibniz-style calculus, but give Newton more credit.
The
college and the city (I think the college was actually first) in
California are named after Bishop Berkeley, but not for any
significant reason (some people liked something he said about
westward expansion).
I
will present Euler’s form of deMoivre’s formula. We will
discuss Euler next time.
I
find it poetic justice that Maclaurin devised Cramer’s rule.
I’ve seen this in original sources (from a past student’s
project).
Simpson
is one of the first examples we see of statistics - we will see
more next time. Realising that errors cancel in means is a
step forward. The is the beginning of the distribution of
sample means.