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.