Mathematics 153
Notes from May 20, 2020, lecture session

What was covered:
The remaining material from Unit 9 discussing the work of F. of Viète and S. Stevin, the comments in Unit 10 (which is exceptionally short), and the material from Unit 11 up to the discussion of coordinate geometry.       
Relevant directory files for Units 9, 10 and 11:
Here is a listing of files from Units 9, 10 and 11 which are related to the May 20 lecture, with comments on their significance.  These links are all clickable. 
http://math.ucr.edu/~res/math153-2020/week8/unit09/history09.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/week8/unit09/history09a.pdf   An
http://math.ucr.edu/~res/math153-2020/week7/unit08/history08a.pdf   Remarks on the loxodrome curves, which are curves on a sphere such that the angles made with both latitudes and longitudes are unchanging.
http://math.ucr.edu/~res/math153-2020/week7/unit08/loxodromic-spiral.png   Another drawing of a loxodrome curve on the globe, this time with rough sketches of major land masses.  
http://math.ucr.edu/~res/math153-2020/week8/unit09/history09.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/week8/unit09/history09a.pdf   An explanation of the mathematical relationship between constructing a regular 7-sided polygon and finding the root of a cubic polynomial.  This should be understood passively.
http://math.ucr.edu/~res/math153-2020/week8/unit09/history09c.pdf  Explanation of Viète's modification to the classical rules, allowing a movable straightedge with two marked points.
http://math.ucr.edu/~res/math153-2020/week8/unit09/neusis-geometry.pdf  Examples of constructions described in the previous file, including angle trisection and cube duplication; however, one still cannot square the circle even if this new construction rule is adopted.
http://math.ucr.edu/~res/math153-2020/week8/unit09/neusis-algebra.pdf  A detailed algebraic discussion of Viète's construction rule.  The difference with classical rules can be stated simply in algebraic terms:  Algebraically, the classical methods allow one to add, subtract, multiply, divide and solve quadratic equations by geometric means.  The added rule also allows one to solve cubic and quartic equations geometrically.  This material is written at the level of Mathematics 172 or a graduate level course in algebra, and it is included only for reference purposes.
http://math.ucr.edu/~res/math153-2020/week8/unit11/history10.pdf  This turned out to be much shorter than originally expected, and it consists of some comments on the scientific method and two scholars who played a major role in its development:  Francis Bacon and René Descartes.  Numerous links for further intormation are included. 
http://math.ucr.edu/~res/math153-2020/week8/unit11/history11.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/week8/unit11/history11b.pdf  This file contains worked out examples involving logarithms and coordinate geometry (only the first type of exercises were covered in the lecture).
http://math.ucr.edu/~res/math153-2020/week8/unit11/log-examples.pdf  More examples of computations with logarithms.  Before the widespread appearance of affordable electronic calculating devices, these sorts of examples were crucial to streamlining arithmetic computations, allowing one to replace multiplication problems with addition problems which are much easier to complete.   These examples should be understood well enough to work others of the same type.
http://math.ucr.edu/~res/math153-2020/week8/unit11/log-2-and-googol.png  A modified version of a whiteboard file from the lecture.  The main changes were that the value of N was computed more precisely and the statement was corrected, with 100 replaced by 101 digits.  This is necessary because a number X with exactly 100 digits is at least 1099 but strictly smaller than 10100.  Of course the word "googol" in the file name refers to a standard term denoting 10100 (not surprisingly, related to the name of the well known search engine).  This example should be well understood.
https://www.youtube.com/watch?v=rJKmc4PVdh4   A fairly old film with instructions for using a slide rule.  The quality is not great by today's standards, but the directions are clear and complete.
http://math.ucr.edu/~res/math153-2020/week8/unit11/sliderule-proof   A brief explanation of why the use of a slide rule's C and D scales actually gives the right answer.
https://www.youtube.com/watch?v=NwW3UHvzcOc   A YouTube video illustrating the use of a (relatively) high precision measurement device called the Vernier calipers, which was invented in the early 17th century by  P. Vernier  (1584 – 1638).
http://math.ucr.edu/~res/math153-2020/week8/unit11/vernier-proof   The algebra which underlies the Vernier caliper, including proofs that the procedure in the video actually gives a correct result.