Remember:  Research topic is due Friday.  Please describe it in a paragraph.  If it is at K-12 level, please include specific reference to standards.  

Quick answers -2.2.2

I think you're now starting to see why this is learning the mathematics that was happening at the same time as what learned about in humanities.  And we'll come and go with that, depending on what you discussed in humanities.  I expect for most of you when we do Asia and the Islamic Empire we should be away from your humanities materials.  Ultimately, I haven't had the opportunity to teach humanities, so you all know more about that than I do.  

Like all history, the simple story is almost never the truth.  Our calendar comes from many sources.  A step was made by Julius Caesar, but neither the first nor the last - there were influences before and after.  This was the first introduction of the leap year (and they had some problems with it).

Apollonius was discussed in class last Friday and is on p. 33 in the book.  

"Hipparchus seemed to be more of an astrologer than a pure mathematician."  Interestingly, the world "mathematician" was taken to mean astrologer for many centuries.  Be wary of this if you find any biblical references casting out mathematicians.  Also, please everyone attend to the important differences between astrology and astronomy.  I can't tell if you're interchanging the very different words.  

It is definitely not the case that Archimedes was the first to prove things.  Surely Euclid did - we have his proofs.  And I think we're pretty sure that *someone* in the Pythagoreans did.  Some say Thales did, some say he didn't, there's the gap in knowledge.  But surely, long before Archimedes.  

Whereas there are things that the Greco-Romans knew that we don't know they knew, it seems highly unlikely to me that there was something they knew that we don't know today.  We definitely don't have complete records, and there are surely achievements we are unaware of, but I do believe there is nothing that hasn't been rediscovered since then.  As we move forward in history, I won't be as sure about this claim.  

Are the Greco-Roman proofs accepted today?  Mostly.  And for those that aren't, it's merely because standards have changed.  I don't think there are any where the result itself is questioned.  

To Archimedes an Octad was 10^8.  It was as high as the greek numbers went at that point (greeks used a system similar to the egyptians with letters for groupings).   Archimedes extended the system to (10^8)^(10^8), the second octad, and so on.  He did this several times octad^octad^octad^octad to estimate the number of particles in the universe as known to the greeks at that point.  Notice this was written to a son of a friend.  Probably just an exercise in demonstrating that numbers can be arbitrarily large.  And the connection to Cantor is probably to power sets, not the Cantor Set.

Archimedes volume and surface area formula derivations are definitely extant.  I think I will choose to not present them, but I will put some on our course website so you can see them.   Similarly, his work on hydrostatics is known.  A quick search for Archimedes and hydrostatics will produce answers to that.  It's a little afield for us, and I'll keep focused on the more mathematics side of things.  

Archimedes war machines are listed in the paragraph that spans pages 37-38.  

The Romans seemed to have a way of wanting to change things to be their way just for the sake of doing so.  I'm not sure that there are any real advantages of the Roman numeration system over the Greek.  

The Greco-Romans made many advances in astronomy.  One might consider that they probably lived in a climate where they spent much time outdoors looking at the night sky.  Also, please remember that astronomy in the quadrivium is a branch of mathematics, which is part of why we focus on it so much.  

We will not talk in class about how star magnitudes are measured.

I do not know what Hannibal did at Cannae - but I imagine it could be easily found.  

We will say something about Nicomachus when we get there (Wednesday).  Same goes for Heron.  But, when Heron used 22/7 for pi, he's merely using one of Archimedes bounds.  It isn't too bad.  

Did Archimedes write all of those books: The Sand Reckoner, Quadrature
of a Parabloa, On Spirals, The Method?  Yes.

If Archimedes is trying to determine the *volume* of Hieron's crown, it probably is of no consequence what it is made of.  

Apparently Johannes Myronas considered the vellum that Archimedes's work was written upon to be more valuable than Archimedes's work.  One possible view of this - what if they had plenty of copies of Archimedes's work around at that point?   _The Method_ is the formerly lost work of Archimedes.  

The isoperimetry problem motivated much of calculus, but doesn't require calculus (although that's now the easiest way to address it).  A special case is the question of whether a square, two squares, or a square and a circle, enclose the greatest area.  Surely Dido wasn't the first to ask questions like these, but it is almost impossible to determine who the first person to ask any question is.  

It is difficult to do mathematics while there are wars and conflicts going on around you.  If you have to struggle to survive, you don't get time to think about things.  Hence the sparsity of mathematics after Archimedes.  

Remember that pi is defined to be the ratio of the circumference to the diameter of a circle.  Polygons with large numbers of sides are close to circles, so can be used to approximate pi by finding their perimeters.  

Archimedes was wanted alive as he was a prominent figure and would therefore make a valuable display.  

Romans maybe seemed more concerned with practical matters (and conquering the world) than their Greek counterparts.  Maybe this is partly responsible for their comparative neglect of mathematics.  

There is a procession of the sun's position of spring equinox so that it changes continuously and returns to where it was in about 25,000 years.  So, next year at this time (exactly) the sun will be at a slightly different place.  And the year after that it will be in another slightly different place, but in about 25,000 years it will be back where it started.  

I'm guessing (but only guessing) that Archimedes was from a relatively well-off family, but that he strengthened his position in the government by all of his work.  

In Hannibal's pursuits against Rome, he wanted to split the allegiances and to get allies from the Roman confederation, but he had to demonstrate that he stood a chance against the rest of Rome because "no one would abandon Rome to join the losing side."  

Hipparchus (and the rest of the greeks) definitely did not have telescopes.  Everything was done with naked-eye observations.  

If you want to know why Caesar was assassinated, read Shakespeare.  

We will talk about how chords relate to sine, though perhaps on Wednesday.  

Archimedes work with lower dimensional cross sections is similar to how we use infinitely thin rectangles in Riemann sums to find area and infinitely thin slices to find voluem.  In spite of all this, Archimedes surely wasn't the first to consider dimension.  If nothing else, it is apparent in Euclid's work, and probably long before.