Feedback notes:  Thank you for the kind feedback.  In particular, I'm glad the reading reactions are working well for many of you.  I think they work well for the class and allow me to incorporate your ideas into the course.  I hope that you are reading the quick answers to see the answers that I don't happen to read in class.

I will not present more history during class for 3 reasons - 1. most people think there's too much history already, 2. this is a math class, in the end, not a history class, and 3. I don't know much more history than what's in Suzuki.  

One more time - exams will be several questions like "compare egypt and babylon" "compare Greek, Chinese and India" and include your own examples.  So, choose your own examples so that you are equipped to discuss the regions and periods that we discuss.  I will *not* hold you responsible for any particular details.  Please stop worrying about this.

Also, perhaps reread a sentence or two before asking a question about it.  Sometimes I skip questions completely when the answer is right there in the reading.  

Quick Answers §2.2.3-

Please remember that mathematics is frequently studied because it's interesting - not just by people now, but throughout history.  Please let go of needing for there to be a use for everything.  

Neither the Greeks, the Romans, or the Egyptians used place value.  The only system we have studied so far that did was the Babylonians.  The Greeks, Romans and Egyptians all used groupings of ten, but they all had a different symbol for ten than for one.  Why are Roman numerals still used today (esp. with all their flaws [ever try doing arithmetic with Roman numerals?])?  This shows the effect of power and force sometimes is more significant than the effect of a system being better.  Whether it was better or not, Roman culture has a significant influence over western culture as a whole.  

Ptolemy didn't discern the angle measurement in his construction for a regular pentagon by using compass and straightedge.  He did that by geometric computation.  He constructed the figure that was as desired by using compass and straightedge.  His accomplishment with the regular polygon was only a shorter way of doing what Euclid had already done.  Not as significant as his other accomplishments.  

I guess we should leave the cause of the decline of the Greco-Roman culture to thie historians, but it is a fascinating question and seems to be a distinctive event in history.  

Hypatia was Orestes "friend and supporter" but was pagan, and so killing her looked good for Christians.  

For Diophantus a pure number is a constant in a polynomial expression.  

Little is known about Hypatia aside from what is written in the text.  We only know what records tell us - and she is the first woman mathematician we have distinct record of.  Were there others?  No way of telling for sure.  

Menelaus was not the first to consider spherical geometry.  It was considered for many astronomical purposes long before that.  

Due to there being so much mathematics in this days reading, I will not be discussing tuning further.  Ptolemy merely said that 5:4 is a much nicer ratio than 81:64, so he used it instead and follow the consequences of it.  Practically the two ratios are so close to not be very discernable.  

I do not believe we have any record of people working for an extended period of time with fundamentally wrong mathematics (not including approximations that weren't as good as they could be).

Metrica is not a person.  It is a work written by Heron - this is why it is in italics.  In the work some theorems are proven, some are discussed and some are merely stated.  The last is what he means by "attributed".  

Like everything else - the path from Diophantus to our modern notation is not a straight one.  

I'm guessing math. teachers were well paid because of respect for knowledge, not because of the usefulness of the mathematics.  

The text says rule 2.2 "is derived using approximations from a table of chords".

As I mentioned before, Archimedes approximation of pi was between 3 1/7 and 3 10/71.  Therefore 22/7 or 223/71 were the best approximations around.  

"why was little progress made in the beginning of the first century?"  Given any 50 (or less) year interval, it's not easy to find particular progress in the ancient world.  I think this is likely merely a matter of incomplete records, if that.  

Building bridges is not making progress in mathematics.  

Diocletian made himself a god - that was his religion.  He banned mathematics in the sense of astrology and numerology, not in the sense in which we do mathematics.  

Probability is a late development in history.  We'll see it eventually.  

Yes, the Augustine on p. 51 is the "St. Augustine" from humanities.  

Books are missing because they are destroyed - many in the destruction of the library at Alexandria - but others in other contexts.  We know how many were there because we have people saying "In these sources I read …"

A solid of revolution is the things you found volumes of in Calc I.




Class notes:

Heron - formula - proof

Nicomachus - figurate numbers
    odd
    even
    square
    prime
    composite
    triangular
    pentagonal
    hexagonal
    pyramidal
    tetrahedral and even others more fantastic than these
    and many relationships
    
Trigonometry
    Hipparchus - chords (and connections to sine) (did use degrees, minutes, seconds, thirds &c as inherited from Babylonian base 60 - and he started with a circle of radius 120 (as a number that worked well with the angle measures - remember we can set any unit we choose for measures) and divided it into minutes, seconds &c, too.  He used the same fraction system for both length and angle)  
    Menelaus - spherical (astronomy) - attempt to be spherical analogue to Euclid    
        sum of angles > 2 right
        AAA conguence
        proves the spherical version of "Menelaus's Theorem" (but doesn't prove the plane version) (product is equal - factors aren't necessarily)
        law of sines on the sphere
    Ptolemy - Almagest (astronomy) [uses aristotle to refuse heliocentric hypothesis of aristarchus] spherical / planar trigonometry
    calculation of chords (builds on Hipparchus)
        36°, 72° using construction for a regular pentagon / decagon
        60°, 90°, 120° using hexagon, square triangle
        can find chords of supplements using equivalent relationship to sin^2 + cos^2 = 1 (which is the same as finding sines of the complement, which is what cosine means).
        Ptolemy's theorem - inscribed quadrilateral
        use it to find chord differences
        can also find chords of half angles
        can also add chords.
        all of this suffices for computing a quite extensive table of chords.  Down to steps of 3/4° ((72-60)/16), but he wants 1/2°, so he approximates.  Makes first extensive trig table.
        Spherical side - uses for astronomy - includes spherical pythagorean theorem (i.e. how sides are related in a right spherical triangle)
        

Diophantus - number theory - notable symbols - only words or lengths before (but for only one unknown and not clear the connection with the unknown and several symbols).  Very notable for considering higher than 3rd power - as it has no geometric significance.  

Finding **rational** solutions to indeterminant equations:
    x+y -5 =0
    y^2 = Ax^2 + Bx + C
    {y^2 = Ax^2 + Bx + C / z^2 = Dx^2 + Ex + F}
    y^2 = Ax^3 + Bx^2 + Cx + d^2
(I don't know how he indicated the different variables which are naturally necessary here - I do know this was all done with special cases and the parameters were merely chosen one by one for examples)

Pappus -

hereafter commentary, but little results of note - the great Greek mathematics fades away as it seems less and less is understood.  Commentary to try to understand then, serves as record and translations for us now.

Hypatia / Theon

Proclus

Eudemus

Geminus

Eutocius

Boethius