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390 Quick Answers 7 March

Your paper draft is due 7 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_ 鈥渟tarting鈥� the paper. You are producing a full paper at this stage. Word count does not include: pictures, equations, or bibliography. Make sure you鈥檙e well clear of the minimum. Also beware: with a missing week between now and then and the exam included, 7 April will rapidly approach. Here are some more comments: /~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 7 April in the afternoon, I hope to finish by early May. I'm a mathematician - so I'm slow at reading papers. Important reminder: this is _ENTIRELY_ a history paper. DO NOT write about now, do not write about teaching do not write about anything but the history. Focus focus focus. This is a short paper. Short papers are not overviews. Focus.

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

There is one class day after today before your midterm exam. _Please_ have a plan. And please come talk to me about it in office hours, that would be great. 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? Here's a fun question - what are the most dull topics?

"Something that I am a little confused about is the exam. I know one of the options is you can write your own topic on the exam... meaning I can guarantee that I will be able to write about one of the things I want to write about during the exam. You had mentioned that we can write outlines and run them by you, but having these topics written out in the outline isn't a guarantee that these topics will be asked about on the exam, correct? We have gone over multiple examples that can be written about during class, so does that mean that these will be options on the exam?"

Sometimes I send messages that reactions are disappointing. This happened more today. Look back at the samples, refocus, and pick up the quality. Also, for both reading and lecture, don't skip large pieces.

For what it's worth, I haven't been updating current averages yet, because we mostly only have reactions. I will update that after the midterm and give you a current average that includes project work, reactions, and the exam.

Lecture Reactions

Perspective is about the world that we see and experience, not an idealisation. We see parallel lines converge on a daily basis.

"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 other answer, as we see in Ferrari's work today, is that they were happy with "answers" that were meaningless as numbers. One more comment in that direction - often trying to work with the answers that come out of the original formula, you may try to cube things and similar. This only leads back to the original equation. I did this once by accident.

The method I presented was essentially the one used by all of the Italian algebraists. (y-a/3)^3 = y^3 -a^2y + .. this is all set up so that the term a^2y will cancel. This is not shocking. It is not trial and error, it is intentionally manipulating the algebra. I鈥檓 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鈥檚 work, we鈥檙e transitioning away from all mathematics being so tightly bound to geometry. There is a fascinating question here - comparing al-Khayyami鈥檚 geometric solution and del Ferro鈥檚 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 鈥� definitely interesting to compare the two - neither is great. Another interesting question 鈥� do we get better? Perhaps surprisingly 鈥� not really. At least not that I know.

Reading Reactions

Not much here, hence the overrun from last time. A list of names that didn't seem to do much. I'm struggling to find a mathematical story to elaborate here.

Amusingly, the number of the beast is 616 not 666. It was a long-ago translation error. More people should know this:

Regiomontanus (Johannes M眉ller) 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.   He was taught by Peuerbach.

顿眉谤别谤 and . Oh, I really don't want to say anything about magic squares. I guess I will try.

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鈥檚 about all we can be clear about. We鈥檝e seen p and m used before. (We see some occasionally, but if you want to know more about notation, look for Cajori鈥檚 _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. Are the symbols really a significant advance? Are the ideas more important than the symbols? Something to think about.

Say something about Rheticus. "I'm surprised to hear that in the past they didn't realize Cos(x) = Sin( 90 - x) given that x is in degrees. I know there were trig tables before Rheticus, by Varahamihira, so I'm confused on why no one recognized that pattern. I'm also very surprised trig was never thought of as ratios of sides of right triangles before Rheticus given how it was interpreted through chords on a circle." The ratios are new, yes. The cosine seems either wrong or just a matter of giving it that name. 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.