390 Quick Answers 10 March

This will do it - I'm reminding myself to ask about diversity summit and other pursuits. 

Your paper draft is due 5 April.  This is the biggest single step.  Your draft is writing the paper the best you can before I give you extensive revisions.  It is _not_ “starting” the paper.  You are producing a full paper at this stage.  Word count does not include: pictures, equations, or blbliography.  Make sure you’re well clear of the minimum.  Also beware:  with a missing week between now and then and the exam included, 5 April will rapidly approach.  Here are some more comments:  https://www.geneseo.edu/~johannes/390draft.html  Unlike the other steps, I will grade drafts in the time order I receive them.  If you submit before 1 April, I should be able to to finish by 11 April.  If you submit on 5 April in the afternoon, I hope to finish by early May.  I'm a mathematician - so I'm slow at reading papers.  

I will be submitting GREAT Day talks, because we are one session (so that I can attend and introduce).  

There are two class days after today before your midterm exam.  _Please_ have a plan.  Please remember - no resources during the exam.  Violation of this will result in earning a zero for the exam.  Chapter 6 is the end of exam-relevant content.  More topics?  


Lecture Reactions

I seem to have handled this almost exclusively through individual comments.  This is in part since we only really had two stories. 

"If Tartaglia's method for solving cubic equations is difficult to use unless the numbers involved work nicely to become perfect squares and perfect cubes, how was Tartaglia able to use this method to solve Fiore's 30 questions? Did the questions use numbers that were easy to work with?"  That's a great question, and we have the contest problems.  Looking carefully at Fior's challenge, they are all "cube and cosa" problems.  These avoid complex numbers and so at least could all be approximated from the formula as much as one wishes by approximating square and cube roots. 

The method I presented was essentially the one used by all of the Italian algebraists.  The first step is simple and should be very clear (y= x-a/3).  I’m not going to re-present the whole cubic solution a third time, but x  = (u+v), and then x^3 = cx + d becomes (u+v)^3 = c(u+v) + d.  Then by creatively equating coefficients (not required but a way to satisfy the equation) we find a system of two equations and two unknowns that allow for solving for u and v, which are then replaced in x = u+v.    Let me be clear - the method is always _correct_, just often not _useful_.  

Although there was a geometric justification in Cardano’s work, we’re transitioning away from all mathematics being so tightly bound to geometry.  There is a fascinating question here - comparing al-Khayyami’s geometric solution and del Ferro’s algebraic - I wonder which is more useful - a graph with an answer to be measured, or an answer that is algebraically exact but impossible to simplify … definitely interesting to compare the two - neither is great.  Another interesting question — do we get better?  Perhaps surprisingly … not really.  At least not that I know.  


Reading Reactions

Not much here, hence the overrun from last time. 

Amusingly, the number of the beast is 616 not 666.  It was a long-ago translation error.  More people should know this:  https://en.wikipedia.org/wiki/Number_of_the_beast

Regiomontanus (Johannes Müller) was a prodigy and advanced unusually quickly through schooling.  He was also a first to talk about trigonometry without reference to astronomy, hence making it mathematical. 

Dürer and conchoid and epicycloid.  Oh, I really don't want to say anything about magic squares.  I guess I will try.  I don't understand what you don't understand. 

Widman largely used + and - for parts of measurements, similar to 2 lb + 1 oz, or 2 lb - 1 oz, which was not a problem for computing, but for a description (not inequalities).   This is the first known appearance of + and -.  That’s about all we can be clear about.  We’ve seen p and m used before.  (We see some occasionally, but if you want to know more about notation, look for Cajori’s _History of Mathematical Notations_.)  Printing encourages uniformity.  When we all see the same things printed we all learn the same ways to write.  This holds for symbols and writing. 

Say something about Rheticus.  Why using lengths vs. ratios is not very different.  Jeff probably overstates this.  Again, using large numbers is only avoiding the need of accurate small decimals. 

Stifel was neither the first to study arithmetic and geometric sequences, nor the first to call them that.  His early steps to compare them together were progress.   And, yes, his steps backward in that sequence were progress for negatives, fractions, and negative exponents.