"One must regard nature reasonably and naturally as one would the truth, and be contented only with a representation of it which errs to the smallest possible extent." -- John Bolyai, 19th century Hungarian
Kline begins:
"Toward the ends of their lives, Euler, D'Alembert, and Lagrange agreed that the realm of mathematical ideas had been practically exhausted and that no new great minds were appearing on the mathematical horizons. Of course, these men had grown old and their vision was already dimmed, for Laplace, Legendre, and Fourier were already in young manhood."
But the hero of this chapter is Karl Friedrich Gauss, who needs no introduction. Here is a little of the great mathematician's bio as described by Kline:
"Karl Friedrich Gauss's brilliance was noted by his elementary school teachers, and they helped him to secure a good education. [No, Kline says nothing about adding up numbers 1 to 100 -- dw.] There is a story that at Gottingen he approached one of his university teachers, A.G. Kastner, with a proof that the 17-sided regular polygon is constructible with straightedge and compass -- at that time one of the outstanding construction problems. Of course, [Kastner] knew that the problem was of theoretical interest, but he did not believe Gauss could solve it. Then Gauss explained that he had reduced the solution of the seventeenth degree equation [that appears in a proof -- dw] to one of lower degree and then solved the latter. For this rebuff, Gauss repaid Kastner, who prided on being something of a poet, by lauding him as the best mathematician among poets and the best poet among mathematicians."
As we know now, Gauss was correct in that he really did construct a regular 17-gon. But no, neither the PARCC, nor any other Common Core test, will ever require high school students to know the steps to construct a regular heptadecagon, as it is very complicated. After this discovery, Gauss was convinced to spend the rest of his life studying mathematics.
And of course, one of his discoveries was non-Euclidean geometry -- one in which the Parallel Postulate is replaced with another axiom. Indeed, Gauss independently discovered the new geometry with two other mathematicians -- one of whom uttered the quote at the start of this chapter, Bolyai.
Kline writes:
"Their chief idea was the one we have already mentioned, namely, that Euclid's parallel axiom is an independent assumption about parallel lines and hence it is logically possible, whether or not it serves any scientific or practical purpose, to replace it by a contradictory axiom. What alternative axiom did these men adopt? One alternative would be to assume that every line through P met l; that is, there are no lines through P which are parallel to l. ...[T]hey had found that theorems deduced from this axiom and the other nine axioms of Euclid contradicted each other. This outcome meant that such an alternative to Euclid's parallel could not be entertained, since a body of inconsistent results certainly made no sense."
First of all, the total of ten axioms (a Parallel Postulate and "the other nine axioms") are mentioned in Lesson 13-6 of the U of Chicago text (five algebraic and five geometric postulates). All the talk about Euclid's "Fifth Postulate" refers to his fifth geometric postulate, not any algebraic postulates.
Anyway, we've discussed this contradiction that Gauss and others found before on the blog. We've seen that a geometry that satisfies all of Euclid's axioms except the Fifth Postulate is known as "neutral geometry." Spherical geometry is a well-known example of a non-Euclidean geometry in which there are no parallel lines. But, just as Kline implies above, spherical geometry is not a neutral geometry, since the Spherical Parallel Postulate contradicts the nine neutral axioms! Therefore the geometry that Gauss and Bolyai discovered is actually hyperbolic geometry -- the non-Euclidean geometry that is neutral, yet a bit harder to visualize than the non-neutral spherical geometry.
In fact, I myself had heard of non-Euclidean geometry, but I never knew that spherical geometry is not neutral until I researched the website of Dr. Franklin Mason for this blog. On an old version of his website. Dr. M used no version of a Parallel Postulate to prove the theorem "Through a point on a line, there is at least one line parallel to the given line." His proof was therefore neutral -- thus demonstrating that parallel lines exist in neutral, but not spherical, geometry.
Kline moves on to discuss the discovery of spherical geometry by Bernhard Riemann, another very famous mathematician. Kline writes:
"Hence, he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded [like a circle -- dw]."
So Riemann had to replace Euclid's Second Postulate in order to make Spherical Geometry work. If Spherical Geometry were neutral, we'd have to keep the Second Postulate (and all others except the Fifth), but again, that's why spherical geometry is not a neutral geometry.
Last summer, I devoted some posts to Adrien-Marie Legendre's Geometry, which includes some work on spherical geometry. Notice that Legendre is one of the mathematicians that Kline mentions at the start of today's chapter as part of the "old" generation. We found out last year that Legendre's treatment has some gaps in it -- first he attempts to "prove" the Parallel Postulate, then much later on he discusses the properties of spherical lines without considering it to be a separate geometry. It wasn't until Riemann -- who was just six years old when Legendre died -- who put spherical geometry on a completely rigorous foundation.
Then last fall, I changed the order of the U of Chicago text so that congruence was taught earlier in the course, well before parallel lines. This meant that I had to pay more attention to which theorems required a Parallel Postulate and which ones didn't (i.e., were neutral). But this neutral geometry didn't connect to the spherical geometry I wrote over the summer, since spherical geometry is not a neutral geometry.
By the way, this summer I plan on returning to spherical geometry on the blog. I am adding the label "Spherical Geometry" to this post as a reminder that I want to finish Legendre this summer, starting at where I left off from last year.
Actually, Kline writes a little about spherical lines in the previous Chapter 25:
"If the surface of the earth is treated as a perfect sphere, and this may be assumed for navigation of the oceans, then the geodesic between two points on the surface is the shorter arc of the great circle passing through these points."
That chapter was all about the calculus of variations. I once took such a course in college, but I never did figure out the integral I was supposed to calculate to prove the geodesy of the great circle. (I couldn't figure out what to do with this strange "arctanh" term!)
In the current chapter, Kline writes more about the better-known results -- that the sum of the angles of a spherical triangle exceeds 180 (for a hyperbolic triangle, it's less than 180) and that in both spherical and hyperbolic geometry, all similar triangles are congruent. (In Common Core parlance, all similarity transformations are isometries.) Kline then describes the experiment in which Gauss tried to calculate the sum of the angles of a large triangle on earth to determine whether geometry was spherical, hyperbolic, or Euclidean. The U of Chicago mentions this experiment in Lesson 5-7, when the students first learn the Triangle-Sum Theorem.
After that long discussion on non-Euclidean geometry, fortunately today's PARCC question is short:
26. Which geometric figures have a measurable quantity?
Select each correct answer.
A. line
B. angle
C. point
D. line segment
E. ray
The correct answers are obviously (B) and (D). We even have two postulates that describe how these are measured -- the Protractor Postulate and the Ruler Postulate. The former is called the Angle Measure Postulate, and the latter part of the Point-Line-Plane Postulate, in the U of Chicago text. The common student errors include mistaking "line" for "line segment" and, as usual, marking only one correct answer instead of both.
PARCC Practice EOY Question 26
U of Chicago Correspondence: Lessons 1-7, Postulates and 3-1, Angles and Their Measures
Key Postulates: Point Line-Plane Postulate
(d) Distance assumption: On a number line, there is a unique distance between two points.
Angle Measure Postulate
(a) Unique measure assumption: Each angle has a unique measure from 0 to 180 degrees.
Common Core Standard:
CCSS.MATH.CONTENT.HSG.CO.A.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Commentary: Since spherical geometry is on my mind today, let's try answering today's PARCC question in spherical geometry. Now a line is measurable since it has the same measure as a great circle. Some authors define a ray in spherical geometry to be half of a great circle, so it also has a measure. Only a point has zero measure in spherical geometry.
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