§4.3 Quick Answers

In §4.2 (which we're finishing today), Abu'l Wafa's proof of the pythagorean theorem is mentioned.  It is *not* (contrary to Suzuki) a dissection proof.  It is a rather simple algebraic one.  I'll quickly say something about both. 

What caused a decline in Islamic mathematics?  Perhaps lack of government support and initiative.  Notice we hear less about leaders pursuing knowledge than we did before.  On the other hand, remember that the early church had banned pagan learning, thus effectively banishing Greek mathematics.  

A very good question - did Khayyami and Nasir al-Din (and al-Haytham) think they had proven the 5th postulate?  I believe they probably did, but I could be mistaken.

I will say a little about al-Kashi's method, but probably not much - it's almost identical to the Chinese method that we discussed in some depth.  He mentions this "least important' work to make this connection back to China.  

I agree, and expect you all will, that al-Kashi's precision is astounding for that age.  We've said this before, but this time I think it really is stunning.  These trigonometry results are most probably his greatest work.  At some point (this surely achieves it) precision becomes something you do to demonstrate your technical prowess rather than for practical uses.  

Commentaries are on the mathematics, not the individuals.  They fill in the details with explanations.  I have ones of Euclid and Lilavati if you want to see examples.  

Abu'l Wafa had a proof of the law of sines.  He may have been the first.

I think by this point, al-Samaw'al had completely arithemticised algebra (i.e. there was no remaining reference to geometry).  This is a significant development.  

Many asked about al-Samaw'al and subtraction.  Here is a translation of his statements:  "If we subtract an additive number from a subtractive number, the remainder is their subtractive sum; if we subtract a subtractive number from a greater subtractive number, the result is their subtractive difference; if the number from which one subtracts is smaller than the number subtracted, the result is their additive difference."

I would completely expect that al-Samaw'al knew that his division did sometimes not terminate.  (Please remember that nonterminating and irrational are not the same thing.)  The notation used in the text for al-Samaw'al's division is pretty much what he used.  Probably he had different notation for the powers, but all else would be the same.  We've come a long way in notation.   

Shah-Nameh Ja'far miniature painting.

Twelvers are described in the paragraph above Nasir al-Din on p. 106.  They support the belief that there would only be twelve imams, they are not necessarily one of the imams.  Before Nasir al-Din, trigonometry wasn't gathered together as one subject for its own sake.  It was more scattered and frequently merely part of astronomy.  

Minutes are subdivisions of degrees, not to replace degrees.  

Asia minor is basically Turkey.  

Please remember pi is **defined** to be circumference / diameter.  There is nothing else it means.