I am about half-done with annotated bibliography comments.  Do you want me to return them when I get them done or wait until they are all done?  I promise I will finish them all by next Wednesday.  

Project comments:  first of all, the more narrow your project the better.  If you think "I don't have enough to discuss", then find more about the topic, do *not* broaden the topic.  If you think you have lots of resources, then narrow your topic.  

Your next task in this sequence is the most important.  For April 5 you are writing you paper.  Write it completely and fully to the best of your ability.  The only sense in which it is a "draft" is that you won't have my comments yet to improve it.  Instead of 5-8 pages, to avoid formatting and font questions, let's make it 1200-2000 words.  (contrary to popular culture, a picture is *not* worth a thousand words, neither is an equation - they don't count toward the word total).  I recommend each of you taking your draft to the writing centre before bringing it to me, so that you can get a second opinion on writing issues.  

Here's some important points.

§6.1

Here's my new theory:  one cause of the renaissance is the plague itself.  Before this time European culture was focused in Christian mysticism and a devalued approach to pragmatic matters.  Through the plague it was shown that not having concern for the world around them led to disaster, so people increased their concern for understanding how things actually worked.  This is only my idea, I don't know if there's any validity to it, but it sounds reasonable to me.  

We will be discussing one-point linear perspective.  There are other perspective systems for more complicated situations.  They were devised at around the same time in a flurry of mathematical art activity.  

I agree that "Alberti was renown for his ability to hold a heavy weight at arm's length for several minutes." may be the most bizarre thing Suzuki has included (could it somehow be  helpful for painting?

Quick research seems to show that Suzuki has a strange typo in his footnote, and there is really nothing about thirty with regard to quarantine - 40 days.  

There's a line between mathematicians and not.  Suzuki only bolds mathematicians.  His decision that da Vinci didn't contribute significantly to mathematics seems reasonable to me.  

It seems reasonable that this is about the time when astronomy and music began being considered as separate from mathematics.  Remember, the first liberal arts were all mathematics (quadrivium), then the trivium was added.  

Pacioli's bookkeeping is a step toward the acceptance of negative numbers.  

Since I didn't before, I should say something about the golden ratio.  

All questions regarding the specifics of cubics and quartics will be addressed on Friday.  

Mathematics is nifty in that people can figure things out.  So, anyone has a potential for figuring something out that no one else has before.  Credentials are worthless compared to ability and insight.  Tartaglia (= Niccolo Fontana) is a good example of this.  There are many examples of mathematical mythologies.  Stories that are told about mathematics.  I think, however, that most of the cubic controversy is actually documented history.  This is definitely an exciting story - and some people's favourite episode in the history of mathematics.  

Pacioli did indeed claim that it was not known how to solve cubics.  Here's a quote from the Summa (not the part in the excerpt):

At these times, it was common for mathematical positions to be decided by contests.  Therefore, sometimes it is best to keep your own methods and not let others know.  Show of solving problems whenever you can, but don't tell people how you do it.  How is this today?  Well, we don't have contests for our jobs, but there is sometimes competition to get new ideas published first.  In mathematics, like in everything else, there are some people who are more competitive and there are others who are more motivated for the common good.  

The revival in Greek included increased interest in all things Greek, thus including Greek mathematics.  

What route is more direct to India than around Africa?  Remember there is surely no suez canal.  There is the land route, but frequently sea routes can be quicker and easier.  

"Cosa" is italian for "thing" which is just as good as "unknown".

The situation with complex numbers is quite different from irrational numbers.  The pythagoreans had a mystical world-view based around the idea that all can be expressed in whole number ratios.  The discovery that a practical length could not be significantly damaged that perspective.  On the other hand, complex numbers were dismissed for years as non-existent and useless.  It was surely a surprise to find that they could have practical uses, but it wasn't damaging to any particular philosophy.  

Latin was spoken in ancient Rome, not anymore by the time of the Renaissance.  

Single entry (like checkbook - but in real time, therefore you can *not* have negative totals) vs. double entry bookkeeping.  (Separate credit and debit books to ensure accuracy.)  

Before Hindu-Arabic numerals it was, indeed, difficult to do arithmetic.  And the system is still in it's newish stages in Europe.  So, surprising it as may seem, most people couldn't do what we now call simple arithmetic as recently as 1400.  

As far as I can find them, I don't see any portraits of mathematicians in Suzuki.  I will be providing as many as I can for you in class (and remember the link on our site to a collection of them).  

In Suzuki's discussion of Cardano's work with 5 + sqrt(-15) and 5 - sqrt(-15), the words "Putting aside" and "mental tortures" are quotes from Cardano on the subject.