§4.2 Quick Answers

We're not in India anymore, and haven't been since chapter 3.  The role of Islamic mathematics has only been more well understood in the past 60 or so years as academics have been taking a more worldwide perspective.  Ethnocentric bias has long been responsible for ignoring important works in other cultures.  In hindsight, the transition (from Greco-Roman to Renaissance) doesn't make much sense without the Islamic story.  Numerals in Islamic mathematics are now being used in an almost completely modern way.  

Remember the ending of the Greco-Roman culture featured less and less understanding.  So, the last copies of the old Greco-Roman works were pretty poorly made.    Also remember that our historical record clearly doesn't have everything we could want in it.  Sometimes we only have someone listing books, but not the actual books, so we only know there was a book once.  

Conic sections go back to ancient Greece.  We discussed them then.  Here's a modern view.  I do not vouch for the link.  It's easy to find others.  Why are they so recurring?  They are the most elementary nonlinear curves and therefore occur more often.  I'm not sure why they are neglected more in classes now than they have been.  But, I do know that all students see each of them (circles, ellipses, parabolas and hyerbolas) at one time or another.  For some reason current algebra instruction has moved away from classifying and analysing them.  

If sexigesimal does not seem practical, why do we still use 360°, and 60 minutes and 60 seconds?  Some things stay because of tradition.  People are reluctant to change things, even when there are better systems.  Decimal fractions are what you call decimals.  

I have said all I will about ibn Turn. Abu Kamil, and al-Uqlidisi.  Likewise, I will hold off on details about al-Khayyami until Wednesday.  

For 301 students, you have heard (or will hear) me talk about co-functions and tangent and secant.  Tangent measures a length of a tangent segment to the unit circle, secant measures the length of a secant segment to the unit circle, and the co-functions are all functions of the complementary angle.  Unlike sine, these terms are all logical.  

Remember that *square* ^2 and *cube* meant actual geometric things.   This is why Diophantus regards powers as square-square and square-cube.  This is why Suzuki makes a big deal about referring to the fourth power.  

I'm not saying much about the abacus here, as it seems away from the main content of discussion.  It does require skill and practice to use efficiently, not merely an abacus.  They can be used for computing roots, and other sophisticated computations.  

Islamic mathematics has a curious interaction with negatives.  It allows them in some places (even as far back as al-Khwarismi), but not in others.  On the other hand, so do we.  It's almost as if they could operate with negatives, but usually chose not to.  They certainly had all the information they needed, and used it in other contexts, but didn't use it as solutions to equations.  

I don't know any more details about Abu Kamil's pentagon.  I do guess, though, you could do it with modern computations and not terribly much work - let a side be x and work with as many pythagorean theorems as you can.  

If AGB is an ellipse in ibn Sinan's construction, the resulting curve would not be a hyperbola.  It would be close, but wouldn't satisfy the defining characteristics.  

There's no finding or proving for tangent, secant, or cosecant.  Merely a choice to compute such things.  

As far as I know, abu Kamil is the first for square roots of square roots, at least as a solution to a problem.    I'm not confident about this.  

I presented ibn Turk's geometric solution to the quadratic on Friday.  Figure 4.2 shows, as labeled, Abu Kamil's pentagon.  Not ibn Turk's conditions needed for x^2 + p = qx to be solved.

Where are these original sources kept?  In libraries and museums around the world.  

The dust boards were not evidence against mental mathematics, but they were evidence against paper mathematics.  

We will see when we return to Europe how Islamic mathematics is introduced.  

As far as I know, the only significance of equilateral pentagons in squares is to do it.  

There are surely documents today that are mistranslated or transferred incorrectly, thus leading to confusion.  

al Khazin, about whom nothing else is mentioned, apparently is given credit for solving al-Mahani's x^2(a-x)=b^2c, i.e. rx^2 = s + x^3.

Historical scorn for astrology is mostly similar to modern scorn for the same.  Any business of magical prediction is either useless or easily disproven.