390 Quick Answers 6 February


Your project topics are due by Friday at classtime.

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.”  

“Will we learn more about (person)?”  No, almost never do we come back to people.  We have all of human history to talk about.  We cannot linger.  We surely return to topics as they are returned to by humans. 

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

I showed the Pythagoreans' proof of the MCRTT in the Elements.  I did not claim it was Euclid's original.  Yes, the MCRTT says that the area of the squares on the legs add to the area of the square on the hypotenuse.  It's a travesty that this is news to you.  Euclid presented SAS earlier in the elements.  It was also surely known to the Pythagoreans. 

Right angles can be trisected, as can any multiple of 3°.  60° cannot. 

The quadratrix is the curve made by the intersection of two moving objects.  The curve is static.  I will (quickly) return to the picture.  Hippias can construct it by plotting points. 

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. 

If you want to know more about Eratosthenes and the circumference of the earth, please watch the little Sagan video I linked. 

Extrasolar objects travel in hyperbolas - objects that pass _thru_ the solar-system and then leave.  The astronomy connections of conic sections were not know at the time of the Greeks.  That was much later.  Ellipses are old, and predate Appollonius.  Equations are long away. 

I skipped talking about Euclid's postulates, I should make up for that. 


Reading Reactions

It is fair to presume that Archimedes used Appollonius' work.  It is fair to presume that in general when we're in the same culture and no one is unusually isolated.  Good to think about such things. 

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.  

I will say a little more about Archimedes and volumes.  We will also see his approximation of π.  π the symbol and name wasn't used until the early 18th century.  (I looked that up.) 

I will talk about Hipparchus, but I may forget to say - he surely did _not_ know that the earth-moon distance was not constant. 

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. 

This is a note to remind me to mention the name Sosigenes around the calendar. 

Yes, there are many towns in NYS named after places in Greece and surrounding areas. 

Lost texts we do not have any known copy. 

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).  I will talk about polygonal numbers, which are probably the most commonly mentioned of those of Nicomachus.

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).