Mathematics 153
Notes from May 13, 2020, lecture session

What was covered:
The remaining material from Unit 6Y  and the material from Unit 7 up to the discussion of infinite series in 14th century European mathematics.       
Relevant directory files for Units 6Y and 7:
Here is a listing of files from Units 6Y and 7 which are related to the May 13 lecture, with comments on their significance.  These links are all clickable.   
http://math.ucr.edu/~res/math153-2020/week7/unit06Y/history06Y.pdf  This is the main file for the subunit and should ultimately be read from beginning to end.  After doing so one should read the cited sections of Burton..   
http://math.ucr.edu/~res/math153-2020/week7/unit07/history07.pdf  This is the main file for the unit and should ultimately be read from beginning to end.  After doing so one should read the cited sections of Burton.
http://math.ucr.edu/~res/math153-2020/week7/unit06Y/fibonacci0.png   Definition of the Fibonacci sequence and the first few terms.
http://math.ucr.edu/~res/math153-2020/week7/unit07/history07a.pdf  This gives a proof of the formula for the nth Fibonacci number as a function of n.   You might be asked a question about the formula itself, but there will be no questions about its derivation.
http://math.ucr.edu/~res/math153-2020/week7/unit06Y/fibonacci1.jpg   Derivations for some identities involving Fibonacci numbers.  These should be read and understood because there are several other problems about Fibonacci numbers in the homework.  
http://math.ucr.edu/~res/math153-2020/week7/unit06Y/fibonacci2.jpg   This file gives the formulas relating a Pythagorean triple and a triple of perfect squares which form a finite arithmetic progression.  Some exercises on this material are, or will be, posted.
http://math.ucr.edu/~res/math153-2020/week7/unit07/history07b.pdf  Another look at perfect squares in an arithmetic progression, this time following a variant of  Fibonacci's approach.  
http://math.ucr.edu/~res/math153-2020/week7/unit07/history07c.pdf   A complete solution to the final problem in Fibonacci's Book of Squares; existing copies of the latter end abruptly without finishing the argument and are sketchy in other places.  
http://math.ucr.edu/~res/math153-2020/week7/unit07/history07d.pdf   A discussion of stereograpic projection, which is a geometrically motivated correspondence between the plane and all point of the sphere except the north pole.  The first proof of this result was due to Jordanus Nemoriarus.  
http://math.ucr.edu/~res/math153-2020/week7/unit07/homework07.pdf  Homework files for Unit 7.
Additional comments:
Information mentioned here might be on the final exam:
1.  Although http://math.ucr.edu/~res/math153-2020/week7/unit07/history07.pdf mentions a few political conflicts between Western Europe and the Islamic world, we should note that there were also long periods of peace, which led to extensive trade between the two cultures and the increased cultural exchange, which among other things led to Western Europe rediscovering Greek mathematics and learning about the advances in mathematics which took place in Asia during the Dark Ages in Europe.
2.  Another noteworthy achievement of Oresme was his work on uniformly accelerated motion.  Specifically, he showed that the distance traveled by an object under such motion is equal to the distance traveled by an object with constant velocity equal to the average of the intitial and final velocities.  He did this using his graphical approach to functions; in modern terminology, he viewed the total distance traveled as the definite integral of the velocity and computed this integral geometrically using elementary facts about areas (note that v(t) = v0 + at for uniformly accelerated motion).