M629 Lecture 5

"Lecture" for Week 5

Scheduling

I still think we can finish Greece this week. I've given you a relatively short homework assignment for next Monday (10/6), which I'll be willing to postpone to Wednesday if there's popular demand. The following week we can easily cover the short chapters on the Islamic and Medieval periods, with another assignment due Oct. 13. After that we'll have to stretch things out again, because the Renaissance and Transition chapters are too long to fit together into one week but too short for two, and the Calculus chapter is also too long for one week.

Electronic library access

Last week I went to a WebCT Users Group Workshop. The most useful thing I learned is how you can get access to journals through the TexasAMU library. Suppose, for example, that you want to read the article by Ravitch that I recommended in the first week. TAMU subscribes to The American Scholar (Phi Beta Kappa quarterly), and that journal is available on the Web back through the issue in question. However, if you go directly to that Web site you will probably get a message that you are denied access because your computer is not located at an institution that subscribes to the journal. What to do? Two methods:

On the page http://library.tamu.edu there is a box headed "What is My Portal?" Log in there using your Neo username and password. You should then be able to go to the E-resource page and thence to the journal of your choice without hindrance.
If that doesn't work, or if TAMU has a paper subscription but there is no electronic version (as usual for old issues), try the "DeliverEdocs" service, also accessible from the library pages. When you register for this service, when asked about "status" be sure to check "Distance Education" (NOT "Graduate Student"). This will give you greater privileges -- e.g., you should not get rude messages saying "Why don't you walk over to the library and get it yourself?"

Comments on Egyptian and Babylonian homework

Egypt 2,4,5,8 were routine. There was a divergence of interpretation in #8 over whether fractional loaves were permitted, as opposed to rounding off.

Egypt 11 (teaching by example): There were a number of interesting comments, so I'm going to create a discussion topic where you can post your remarks. Several people will get special invitations to do so.

Babylon 1 routine.

Babylon 3 (sexagesimal multiplication): Some of you didn't really get into the spirit of this one, doing the multiplications in decimal and converting to base 60 only at the end. The prize goes to Alan Lacko for the most throughly Babylonian approach, complete with cuneiform notation (or an ASCII equivalent). I'll ask him to post his solution (from now on such a remark is implicit whenever I mention a particular optimal contribution).

Babylon 7 (false position for 2x2 linear systems): Most of you fell into a trap here. Given ax + by = c with a not equal to b, if you solve az + bz = c and then let x = z + d, y = z - d, the result does not satisfy that equation! So solving the other equation for d accomplishes nothing good. Instead, you have to weight the +/-d terms appropriately with a and b. De-Vonna Clark has the cleanest treatment of this. (Of course, the general solution with letters for the parameters would have mystified the Babylonians....)

Babylon 11 (reducing the cubic): This gave a surprising amount of trouble. Some of you had parameters in the transformed equation that depend on x, which is definitely not allowed. A straightforward analog of "completing the square" removes the x² term; that's what some of you did, and what I thought was involved when I assigned the problem. However, the problem as stated asks us to remove the x¹ term instead. That also can be done by a linear change of variable, but the algebra of finding the right coefficient is more involved (requiring solution of a quadratic, as Allen's text says). Rich Enderton has the best treatment here.

Babylon 12 (linear interpolation): One person asked "What does linear interpolation mean?" and others should have. The most effective method for this problem is "linear inverse interpolation". (Direct interpolation leads to repeated guessing, which is what several of you did.) For the first question, observe that 55 is 19/44 of the way from 36 to 80, two of the numbers in the second column of the table. So we expect that n lies roughly 19/44 of the way between the corresponding numbers in the first column -- i.e., n = 3 19/44 = 3.4318. The exact value is 3.4971. Karen Bump has one of several good answers to this problem.

Later Greek mathematics

I have very little to say here, because I suspect most of you know (or remember) more about Euclidean geometry than I do.

Antiphon, Eudoxus, and Archimedes. In Exercise 5 Allen raises the issue of whether Archimedes deserves to be called the inventor of calculus. That accolade is presumably based on his calculation (really, definition) of the area of circles by means of approximating polygons -- effectively a sort of Riemann sum. Although Archimedes was the first to make this topic quantitative -- thereby dragging Greek geometrical thinking a big step in the modern direction -- Allen points out elsewhere that Eudoxus had earlier made extensive use of this "method of exhaustion" and would even have understood "volumes of solids of revolution by slicing".
It seems worthwhile to note (from Heath, Vol. I, pp. 221-223) an even earlier reference to the method. Antiphon, a contemporary of Socrates, discussed the squaring of the circle this way: Inscribe a square in the circle; on each side of the square, as base, put an isosceles triangle with its vertex on the nearby arc of the circle; on each side of this octagon put an isosceles triangle; etc. According to the commentary of Simplicius, "Antiphon thought that in this way the area [of the circle] would be used up.... And, as we can make a square equal to any polygon ... we shall be in a position to make a square equal to a circle." Aristotle mentioned Antiphon's argument only in order to denounce it as a well-known fallacy, and of course it is a fallacy within the Euclidean ruler-and-compass framework. But Antiphon was clearly ahead of his time! Heath says, "But the objection [that the whole area will never be used up] is really no more than verbal; Euclid [presumably following Eudoxus] uses exactly the same construction in XII.2, only he expresses the conclusion in a different way, saying that, if the process be continued far enough, the small segments left over will be together less than any assigned area. Antiphon in effect said the same thing, which again we express by saying that the circle is the limit of such an inscribed polygon when the number of its sides is indefinitely increased." How close Antiphon, Eudoxus, and Euclid came, not only to Archimedes, but to Leibniz, Cauchy, Riemann, and Weierstrass!