390 Quick Answers 8 March

For those who are here in class today and punctual to all of next class,  you have earned +2 on the final exam.  Thank you for being here on an unpopular day. 

We have an exam next Friday.  Bring computers; you will be typing in class.  Do *NOT* access any materials during class.  Please please have a plan.  I am happy to discuss plans in depth during office hours.  I have an office hour every day between now and then, although I wouldn’t recommend coming on Friday for this just because you won’t have time to make a new plan.  Because the semester started later than in years past, but Spring Break did not move later, I will not have a natural extended time to grade the exams.  As you may imagine, it is quite a task to do so.  Please be patient in my returning scores for them.

I promise to complete reading annotated blbliographies no later than Monday, 11 March. 

April 5 is far sooner than it feels.  Please please please be diligent and dedicated in working on your drafts now.  I am always happy to talk with anyone about it.  When I get done reading exams I will read paper drafts in the order I receive them.  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.  If you turn your paper draft in on 5 April, I may return it to you as late as early May.  Please submit them early.

“Does the bibliography for the final paper (and draft) need to have the annotations, or is a list of the sources cited in the proper format sufficient?”  No annotations when you write a paper.

I'm glad that people are liking the portraits.  I do try to include them whenever I can.  I expect each of them knew and in some sense acknowledged the portraits that we are seeing.  I only didn't before because we don't have any from contemporaries. 

Anyone who _didn't_ use the library survey for this time may use it for next time.  Not beyond.  Thank you very much for those who did.  I appreciate they library and I am grateful for their support.
One more tiny point … there’s nothing stopping you from completing the reactions due next Monday as early as today.

Again, I enjoyed your reactions tonight - thank you.  

Lecture Reactions

Bombelli was the first in history to work with complex numbers.  He did so by trusting that √-1 would behave like other numbers, including squaring to be -1.  It is very natural to say that √-1 does not exist.  It was a big step to think that it could be useful.  Bombelli trusted that complex numbers would work the way other numbers do, and that led to the results and methods.  The point of Bombelli’s work was to find a way to work with the answer as del Ferro’s work produced it. 

Don't feel bad for not having heard about versed sine before, it's not as common as haversine. 

Rusty compass constructions (with a fixed circle size) are limited.  Fewer constructions can be made using this limitation.  There is plenty of research on this topic. 

Someone asked when do we know that the sun moves.  This was an idea of Halley, Cassini and Herschel in the 18th century, verified by Argelander in the 19th century.  [I had to look this up.] 


Reading Reactions


Viete did not fix the calendar, in fact, he was complaining about it being fixed.  Pope Gregory didn't either, but the Gregorian calendar bears his name (like the Julian calendar in that way).  I will finish our calendar story. 

Oh this is a good time to say - yes, coding is definitely mathematics.  The NSA employs more mathematicians than anyone in the world.  They have positions at all levels.  If you are comfortable not telling anyone what you do, I have heard good things from alumni and friends of mine who work there.  It is a curious compromise between academics and industry, as they do spend a significant amount of time learning new mathematics - taking and teaching classes - all about the things that we don’t know about.  

Presumably van Roomen created the problem using powers of sines as Viete solved it - by expanding sin(45ø) using angle addition repeatedly.   There is not another reasonable way to imagine the problem being created. 

Ok, I don't think I appreciated this enough before - yes, Roberval is the first known to consider the sine curve, in it's role as the companion of the cycloid.  Did he connect it with the sine trigonometry?  It seems perhaps not yet. 

Re: Fermat in general - asking questions is valuable, making conjectures is valuable, and proving things is also valuable.  They all have their place.  We need all of them.  

It is a tricky problem to ask who first considered integrals.  Integrals are area problems.  And area problems are as old as history.  Some of them are done by cutting regions into smaller pieces.  That’s old too.  When do they become integrals?  Perhaps when they connect to derivatives, or when they become systematic (which happens the same time).  And that is what calculus is all about.  We are hot on the heels of calculus right now.  We need one more piece at the beginning of 7.2, and then we’ll have full calculus story next week.  Well, at least that part of it.  Also if you’re looking for this, you might remember to look back at Stevin, who had some early insights.  3 main problems of calculus:  derivatives (slopes), integrals (areas), and optimisation.  

I might not have mentioned this without someone asking, but yes, it is very reasonable to expect that Pascal was familiar with other's work on the arithmetic triangle. 

Cardano did work with probability as a gambler before Pascal.

For anyone thinking about computing machines as topic for the final, Pascal is a good place to start.  Pascal's adding machine.