390 Quick Answers 6 February

My reminder takeaway from research weekend - questions are important and even saying "I don't understand" is impressive.

Your project topics are due by Friday at classtime. As of an hour ago I have 4. Given that this one is merely a short paragraph, please paste or type in. That is better than submitting files. In the future, submitting files will be better. For those .pdf is always best. Learn that unless you are sending to someone to edit, .pdf is always best in all settings. It stands for "portable document format" and is designed with the sole purpose of sharing.

These are _quick_ answers, and so I can never explain the mathematics more deeply in them. I know I rush through the mathematics, in part because I know that the details are not the point. This is very important - if you _ever_ want to know more about _anything_ and come and talk to me about it, I promise to say “oh, that’s great, let’s talk about it.”

For your exams, your essays are required to include examples, (a total of 12 for the first exam). And example includes who, what, where, and when.

We're so far away from people wanting credit for work. Forget about it until we start to have disputes about it. You'll see it then.

Lecture Reactions

We jump 1000 years (not quite, but close) because not much of significance is known between ~1400 and ~650 BCE. [Curious, while checking these dates, I saw that Jeff claims the Berlin papyrus is from the 19th dynasty, which would put it around ~1200 BCE, but every other source I see has it much earlier, around ~1800 BCE. I can't find a reason why he would be so far off. I will keep looking.]

Loose recap of Eudoxus - A is proportional to d^2 for polygons, and because circles are arbitrarily indistinguishable from polygons, the same must be true for circles also. The most important point is not that a 16-gon is close to a circle, but that a 2^n-gon is arbitrarily close, i.e. as close as you could ever want, for some n. Indirect proof is identical with proof by contradiction.

Hippias was moving lines in a square, so that the one turning takes as long to turn as the one falling takes to fall. Then plotting points.

The golden ratio was known generally at the time of the Pythagoreans, and perhaps even to the Egyptians.

Euclid gathered the Elements from prior work: Thales, the Pythagoreans, Hippocrates, Eudoxus, … all those who came before. The work on number theory - greatest common divisor, infinitely many primes was his.

Reading Reactions

Elephants came from India, where they were commonly used in battle. Reading Indian history is fascinating. We'll get some next week.

There are many locations in central NY named for Greek originally.

Ok, I say again, Roman numerals are awful, were never used for mathematics and only used for labeling. If you have some misguided nostalgia for Roman numerals, try multiplying MCMXCIX and CDXLIV without converting. The subtraction is really the worst part. This is consistent with the Romans being more interested in conquering and making rules for others than learning. At the colosseum there is an entrance numbered 106 and written CIIIIII (similar examples). The fact that they are still around has more to do with the force of the Roman empire than their utility. The Roman mathematicians mostly used Greek and adapted Babylonian numerals, still working with minute 60ths and the second subdivision. If it's so bad, why do people still use it? See also why the US doesn't switch to metric, and why the government forces ignorant practices on the people.

The Sand Reckoner is _not_ practically realistic, it is merely an excuse to talk about large numbers. Octad notation is groups of 10^8, and is in some ways similar to scientific notation.

Archimedes did not use the symbol π, but did think of it as C/d, as we do. π the symbol and name wasn't used until the early 18th century. (I looked that up.)

We will talk about chord tables. They _are_ trigonometry. They are _not quite_ sine, but they are related. Sine comes later (in about 2 weeks).

This is a note to remind me to mention the name Sosigenes around the calendar. Please give him credit, and not Cesar, which ... clearly he didn't figure this out. I always need to point this out - we do not currently use the Julian calendar. If you don't know this, stick around to see when it's changed and why.

I will draw some pictures of Nicomachus's figurate number. Some of the point of Nicomachus is that he made up names for so many different things. Most are not seriously studied (but you could if you wanted).

In discussion of Ptolemy, and how his astronomical view, while the longest in human history, is no longer accepted, the mathematics remains. Because math. is proven, it continues and is just as valuable as it was then. This is why study of history of mathematics is much more important than history of science. And, why it takes so long to learn mathematics - you can't just start at modern mathematics - you need to learn all that came before. For what it's worth - most of you have made it to the 18th century now in your classes.

At this point in time "mathematician" often meant "astrologer" i.e. "seer" and banning this practice seems entirely reasonable. They were not banning those who studied our course content.

Stay tuned for non-Western mathematics in Chapters 3 (China & India) and 4 (Islamic) before a return to what little is happening in Europe next (Chapter 5).