Math 629: History of Mathematics
Frank Sottile

• Book Review: Please see the information page .
• Opening Remarks: Many believe that mathematics started with numbers; counting, measuring, planning, settling accounts, etc. Without entering that debate (depsite teaching this course, I am not an historian, but rather a practising mathematician), number udoubtedly is both fudamental and old. The earliest civilizations that left records, Egyptians, Babylonians, Chinese, Maya, ..., all left records involving numbers. Some even had privileged classes of scribes who performed clerical duties, including many impressive computations. We will begin our course looking at some of this material.
• Print-vs.-web The availability of electronic resources allows us one to reach a large audience quite easily. This has greatly increased the quantity of information available, but in many cases dramatically decreased the quality of that information. The ancient admonition that the buyer should beware is more true in this age. While I will assign and use on-line resources for this course, we should all keep in mind that they are not as carefully edited as print material. Thus they will contain typographic errors, as well as factual errors, and may have more biases than a typical book.
• Reading:
  • I would like you to read Allen's The Origins of Mathematics, which is about the origins of counting. You should appreciate how difficult it is to delve into this question. I note that this source become less complete nearer to its end, more like talking points in a lecture.
  • Please also read his A General View of Mathematics Before 1000 BCE (BCE:= Before the Common Era, which is the synomym for BC that Historians use). This focusses on Chinese and Indian Mathematics.
  • Read also Allen's Babylonian Mathematics.
  • Finally, read the St. Andrew's pages on Babylonian Mathematics.
  • Note that I am (ahistorically) deferring Egyptian mathematics to next week.


• Assignment: (HW 2) Note that this is based on the reading assignment.
  It is OK to discuss this among yourselves. We all have something to learn from each other.
  To hand in: Email a .pdf to both Mehrzad Monzavi Mehrzad@math.tamu.edu Due Monday 22 January. (If we have a grader by then.)
  1. Write one or two paragraphs answering the following questions: What are the two kinds of numbers? How are they used in common life? What evidence is there that non-Human animals use/understand either type? How about babies or very young children? Which type of number lends itself to arithmetic?
  2. Allen gives the formula (n³+n)/2 for the (common) sum of the rows/columns of a magic square with side length n. Explain why this is the sum of any row/column of a magic square of side length n.
  3. In Allen's text, the Method of the Mean is discussed as a way to approximate the square root of a number n: Start with some initial guess, e.g a=1. Then, replace a by (a + n/a)/2, the average of a and n/a (which together multiply to n). On a calculator (or better) a computer carry out this procedure for a few steps (3-5), for some natural number n that is not a perfect square. Record and hand in the (base ten, digital) steps that you compute. If you have access to a computer and can do this with many digits of precision, by all means use it, and report your answers to higher precision (such as 100 digits). If anyone gets a particularly interesting answer, share it in a post on Piazza. This problem is very relevant to modern computation; I'll explain this after we share our answers.
  4. I'd like us all to practice some calculations using the sexagesimal system of the Mesopotamians. For this, let us use the notation that `;' represents the `sexagesimal point' and `,' is the delimiter between `places'. Thus `2,22' is one-hundred and fourty two, while `1;45' is 1.75. Do these using base 60 and show or explain your work. The purpose of this is to appreciate what it is like to compute in base 60. To that end, do not simply converty to base ten, do the computations, and convert back. Do them purely in base 60, employing the usual algorithms you know.
     1. Warm-up: Express the (decimal) numbers in sexagesimal: 45   150   3253   17589   100,000.
     2. Simpler: 20 + 50 = W     7*17 = X     3,9 - 1,40 = Y     1,24*1,24 = Z.
     3. How about some division: 1/2 = V     1/3 = W     2/5 = X     7/4 = Y     2,16/3 = Z.
     4. Repeating sexagesimals: Why is 1/59 = ;1,1,1,.... ? My favorite decimal fraction is 1/7. What is this in sexagesimal (multiply your answer by 7 to check) ?
     5. Challenge (only if you have stamina and like this stuff; this is not required): Try to verify that (1;24,51,10)^2 is pretty close to 2, as recorded on YBC 7289. I found this interesting.
  5. Explain how the Babylonians used tables of squares (n²) to facilitate multiplication of whole numbers. Compare this to the method we commonly use for calculations by hand. Which do you think makes more sense for sexagesimal calculations? Why?
     Table of Babylonian squares.
  6. While Babylonian mathematics is fascinating—my favorite story is how they solved quadratic equations, so that the familiar quadratic formula is just a reworking of method—I have already assigned enough to keep us busy.