"The spectacle of the universe becomes so much the grander, so much the more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements."
So we see that this chapter is all about minimizing things, including physical laws (which we've discussed in the previous chapter as well). Now Kline begins with a problem familiar to the readers of this blog:
"According to one of the legends of history, Dido, of the Phoenician city of Tyre, ran away from her family to settle on the Mediterranean coast of North Africa. There she bargained for some land and agreed to pay a fixed sum for as much land as could be encompassed by a bull's hide."
Ah yes, we are discussing the Isoperimetric Problem, Lesson 15-8 of the U of Chicago text. Actually, back when I was researching the Isoperimetric Problem for this blog, I saw references to the ancient Queen Dido, but I didn't mention her on the blog at all. Actually, what I knew then about the Dido problem was incomplete -- I didn't know about the following twist until I read it in Kline:
"Her second bright idea was to use this length to bound an area along the sea. Because no hide would be needed along the seashore she could thereby enclose more area."
We know that the solution to the Isometric Problem is the circle -- the curve that encloses the most area for its length. We've also seen questions in which we are to maximize area by building a fence along a river to enclose a rectangular area -- the answer is a rectangle whose width is exactly half of its length. Combining these two ideas, we can solve the Dido problem:
"According to the legend, Dido thought about the problem and discovered that the length of hide should form a semicircle."
So we see that without water, the largest area is a circle -- with water, it's a semicircle. If we restrict to rectangles, without water the largest area is a square -- with water, it's a semi-square (that is, half of a square, or a rectangle whose width is half of its length).
We found out earlier (Chapter 6 of Kline) that the reason for the semi-square involves reflections -- if we reflect the rectangle across the river, we have a larger rectangle whose area is maximized for the new fixed perimeter (double the original fence length). The solution to that problem is a square, so that of the original problem with the pre-image rectangle is a semi-square. Similarly, if we reflect Dido's land across the water, the figure should be a circle (the solution to the Isoperimetric Problem), so the pre-image is a semicircle.
Kline writes that the life of Dido -- perhaps the world's first female mathematician -- ended tragically:
"[Dido's new lover] Aeneas was a man on a mission, and he soon departed to found a new civilization in Rome. Dejected and distraught, Dido could do no more for Aeneas than to throw herself on a blazing pyre so as to help light his way to Italy...Rome made no contributions to mathematics whereas Dido might have."
Kline writes only three equations in this chapter -- all of them involving integrals. In this chapter, he proceeded to write an equation to describe an object falling on an arbitrary curve, which looks very ugly when we try to write the integral in ASCII:
T = 1/sqrt(2*32) int _0 ^x1 (sqrt((1 + y' ^2)/y) dx
This integral is used to solve the Brachistochrone Problem -- find the curve that minimizes the time it takes for an object to fall from P to Q not directly below P. Three Swiss mathematicians worked to solve this problem: the Bernoulli brothers (Johann and Jakob) and later on, a more famous name I've mentioned several times on the blog -- Leonhard Euler. Their work laid the foundation for a new branch of mathematics -- the Calculus of Variations.
Question 25 of the PARCC Practice Test is on constructions:
25. Using a compass and a straightedge, a student constructed a triangle in which
The compass is opened to a set length and two intersecting arcs are drawn above
What could be the set length of the compass so that triangle XYZ is isosceles but not equilateral?
Select all that apply.
A. less than 1/2 (XY)
B. equal to 1/2 (XY)
C. between 1/2 (XY) and (XY)
D. equal to (XY)
E. greater than (XY)
Select the correct phrase to complete the sentence.
If the opening of the compass is Choose... [from the same choices as A-E above -- dw], then triangle XYZ will be equilateral.
What's this? Here's yet another question that's on classical constructions,right after I spent an entire week in an Honors Integrated Math I class studying constructions!
Anyway, Part A of this question is similar to the earlier construction problem, Question 21. We recall how Jericho set his compass to be less than the length of the segment
At this point we should begin to wonder, what does 1/2 (XY) have to do with anything? Well, we can try an actual compass set to less than 1/2 (XY) and see what happens. The result is that the two arcs fail to intersect, and no triangle is formed. The problem here is the Triangle Inequality -- the longest side of the triangle can't be as long as or longer than the the sum of the other two sides. So if one side has length XY, the other two sides must be more than 1/2 (XY). Neither (A) nor (B) produce a triangle (in the latter case, the two arcs intersect right at the midpoint of
Also, there's no reason that the compass setting can't be greater than XY (even though it's always less than the segment length in Question 21). This never violates the Triangle Inequality since the length of the long leg is always less than the sum of the other long leg and the base
For Part B, we want the triangle to be equilateral. Of course we want all three sides to equal XY, so the fourth choice from the drop-down menu -- the same as choice (D) above -- is correct.
I suspect most student errors will occur in Part A. We begin with the usual problem with a multi-part answer, namely that students will only choose one answer even if two or more are correct. But even if the students realize that they must choose more than one answer, they might be influenced by Question 21 to choose answers like (A) or (B).
In fact, come to think of it, the earlier Question 21 is poorly worded. That question doesn't consider the possibility that the compass setting is less than half of PQ -- and if it were, none of the five choices from Question 21 would be constructed! And I admit that I probably would have ignored the less than half case -- except that just four questions later, we're forced to consider that case! Who knows -- perhaps we're supposed to know that the compass setting is more than half of PQ in Question 21 by looking at the diagram, which would be impossible otherwise. We're only given that the setting is less that PQ so that we know not to choose "an equilateral triangle" (or "a 60-degree angle") for that question.
PARCC Practice EOY Question 25
U of Chicago Correspondence: Lesson 3-6, Constructing Perpendiculars
Key Theorem: Construct the perpendicular bisector of a given segment
Common Core Standard:
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Commentary: Inspired by the story of Dido, I decided to add two construction questions to construct a region over which she could rule. I changed "bull hide" to "fence" in order to avoid distracting the students who might wonder what the hide is for.