390 Quick Answers 28 March

I do wish that I could be with you today, and I promise this won't happen again.  Please find the lecture video link immediately below the link you followed to get here.  Also, if you want, there's theme music in the link above, and in some cases a bit more information (especially about maps:  including the formula for producing the Mercator proejction) in the links below than there is in the video.  Read this, watch the video lecture, and write your lecture reactions on all.  Thank you for getting reactions done early - I do still find it odd that that was more successful than having two weeks to do it, but I'm glad for good. 

Remember, the draft of your paper is due a week from Monday.  Remember that I will be returning comments in the order received.  Remember if you submit it a week from Monday, you may get it back in early May.  Thank you for your dedication.  I will talk about them in person, but I will not be reading drafts online and giving feedback before they are due.  I strongly recommend visiting either the writing learning centre or the history writing learning centre.  I will have regularly scheduled office hours next week (ok, no, I won't.  I will have email office hours next Thursday, as I will be administering an exam).  Oh, that's subtle, so ... if you have questions you may email me on Thursday and I _will_ reply, contrary to the above.  

Those presenting at GREAT Day need to schedule two rehearsals with me before your talk.  They may not be in the same week.  They may not be the same week as GREAT Day.  Be careful.

Be very careful in putting the timing together in Chapter 7.  Let’s look at the dates for the three sections.  We’re jumping back and forth.  England isn’t behind … yet, but they are about to be. 


Lecture Reactions

I don't think Stevin was encouraging to always use the notations on the decimal places.  They're like training wheels. 

This is how Descartes is spelled.  He's kinda a big deal.  Oh, and Leibniz is important also, and his name is worth spelling and pronouncing correctly. 

Someone made a nice comparison for Descartes talking about xx instead of x^2.  You write f', f'', but eventually switch to f^(4).  It's about the same.  It's definitely not a big deal.  Why did we switch x and y?  That's a good question, and I have no idea.  Watch for when the switch happens. 

Leibniz's work on calculus were papers, not books.  Both Newton and Leibniz were familiar with the chain rule.  I think Leibniz didn't discuss it, but merely used it by multiplying his differentials.  In that context the chain rule says "dy/dx * dx/dz = dy/dz" and so he probably didn't find it worth special mention.  I think it's fair to say that Leibniz had almost all of Calc I (not including limits, but he did that differently), but not much for Calc II. 

Yes, calculus was done before functions.  Instead they worked with equations, like you do implicit derivatives. 

It's never a bad time to remember - there is more mathematics being created now than ever in history.  We stop in 1950 not because there isn't any more, but because there's way too much. 

I said little about it in class, but someone gave me an opportunity to say more, aside from the machine pictured, Leibniz had a fascinating device that you could trace a curve and it would draw the derivative, or you could use it backwards and it would draw the integral. 

Oh, I remember this shows up now: ± ∓ are used only in pair to mean use either the top of both or the bottom of both.  I also remember that most of you don’t know this, so it’s good to talk about.

For those who could notice a difference, I believe the well-tempered (modern) tuning sounds better largely because it is so much more familiar.  I am on your side on this, but it makes sense.  I think music adopted the idea from mathematicians.  It wasn’t forced upon them.  Yes this is the well-tempered of Bach’s titles which were new at the time.  Why were octaves and fifths thought to match up eventually?  Probably because the system was defined “locally” first, not stretching up the necessary 7 octaves to match up.  The way the system is defined has this logical consequence that just happens to not quite work arithmetically using small whole number ratios.  Playing notes on a piano is merely pressing a key.  Playing notes on a string instrument is about choosing the correct string length.  Choosing the correct string length is easier with small number ratios.  This is the end of our musical topic.  Probably don’t write about this. 


Reading Reactions

I will surely say something about this in the video, but … it is a shame that we use the Mercator projection today.  It is _only_ useful and relevant if someone is traveling by compass across open space.  And, almost no one does that.  The map gives so many false impressions and is not useful for any reasonable purpose.  It would be great if no one saw it ever.   Latitude and longitude are quite very old, going back to ancient Greece.  Cylindrical projection (which I talk about in the video) is 1. older, 2. simpler, 3. area preserving, and generally 4. better, except for compass navigation. 

"When does the general public start to do math?"  Those who have taken INTD 203 know more about this than I do, but it looks to me as if public education began in the 18th century.  I will completely defer if I am wrong. 

Yes, at this time, and for quite a while (I know it in Poisson in 1820), 1/0 = ∞ and 1/∞ = 0.  In fact, since they didn't have a typeset for ∞, they often would use 1/0 in its place. 

Prosthaphaeresis is using trigonometry (tables) to multiply.  It is _not_ used to compute trigonometry.  It converts multiplication of two numbers into addition of two numbers, like logarithms do.  Remember accurate astronomy requires manipulating large numbers with high accuracy.  Would you rather add or multiply ten digit numbers?  But with the invention of logarithms, prosthaphaeresis becomes unnecessarily cumbersome in comparison.  The trig. identity used in prosthaphaeresis was not new, but the idea of using it to simplify multiplication was new.  This is part of the reason that if one wants to check about crediting, it is difficult to do. 

Harriot is a nice place to point to someone who was ok with negative roots.  We’ve been wondering where this would come.  Harriot was ok with complex roots.  The symbol i is due to Euler - yet to come.  Never too early to practice pronouncing Euler.  

Trigonometry functions (and next logarithm functions) were looked up in books.  Mathematicians compute them once and others (including other mathematicians) look them up.  There has never been a time in history when either are computed by hand widely even by mathematicians.  Someone does it once, and everyone looks it up, until we have calculators, which are like a table in a machine.  

Ok, I feel bad about not talking about Wallis more, and a couple people independently found that he created the infinity symbol, and I know you like notation, so … there.  ∞.  Enjoy.  

This is the very beginning of the topic of journals.  This will be another theme going forward.