Abraham Lincoln and Geometry

While re-reading the wonderful parallel biography Team of Rivals: The Political Genius of Abraham Lincoln by Doris Kearns Goodwin, I was reminded of this passage from Lincoln’s time on the Illinois traveling law circuit in the 1850s, the interlude between his term in the House of Representatives and his ascent to the presidency:

Life on the circuit provided Lincoln the time and space he needed to remedy the “want of education” he regretted all his life. During his nights and weekends on the circuit, in the absence of domestic interruptions, he taught himself geometry, carefully working out propositions and theorems until he could proudly claim that he had “nearly mastered the Six-books of Euclid.” His first law partner, John Stuart, recalled that “he read hard works — was philosophical — logical —mathematical — never read generally.”

[Law partner William] Herndon describes finding him one day “so deeply absorbed in study he scarcely looked up when I entered.” Surrounded by “a quantity of blank paper, large heavy sheets, a compass, a rule, numerous pencils, several bottles of ink of various colors, and a profusion of stationery,” Lincoln was apparently “struggling with a calculation of some magnitude, for scattered about were sheet after sheet of paper covered with an unusual array of figures.” When Herndon inquired what he was doing, he announced “that he was trying to solve the difficult problem of squaring the circle.” To this insoluble task posed by the ancients over four thousand years earlier, he devoted “the better part of the succeeding two days… almost to the point of exhaustion.”
Doris Kearns Goodwin, Team of Rivals: The Political Genius of Abraham Lincoln, pages 152-153

I have two thoughts on this: one mathematical, and one political (albeit the politics of the 19th century).

I must admit that I’m charmed by the mental image of Lincoln, like so many amateur (and professional) mathematicians before and after him, deeply engrossed after a hard day’s work by the classical problem of squaring the circle, described by Wikipedia as “the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge.”

A subtle historical detail was left out of the above account, one that I would not expect a popular history book to include. While it’s known today that squaring the circle is impossible, this was not a settled question during Lincoln’s lifetime. Indeed, the impossibility of squaring the circle was settled in 1882, seventeen years after Lincoln’s death, when Ferdinand von Lindemann proved the transcendence of $\pi$ — that $\pi$ is not a root of any polynomial with integer coefficients. All this to say, when Lincoln spent two days attempting to square a circle, he was actually working on a celebrated open problem in mathematics that was easily understood by amateur mathematicians of the day… in much the same way that the Twin Prime Conjecture attracts attention today.

(As a personal aside: I still remember the triumph I felt a student many, many years ago when I read through this proof in Field Theory and Its Classical Problems and understood it well enough to stand at the chalkboard for the better part of an hour to present it to my teacher.)

Politically, I was reminded of the wonderful book Abraham Lincoln and The Structure of Reason by David Hirsch and Dan Van Haften. Hirsch and Van Haften argue that Lincoln’s studies of geometry were not merely for idle leisure or personal satisfaction, in the same way that people recreationally solve crossword puzzles today. Instead, they argue that Lincoln’s penchant for persuasive rhetoric was shaped (pardon the pun) by his study of geometry, and that Lincoln’s speeches tended to follow the same six-part outline that Euclid employed when writing geometric proofs in The Elements.

Square roots and logarithms without a calculator (Part 12)

I recently came across the following computational trick: to estimate $\sqrt{b}$ , use

$\sqrt{b} \approx \displaystyle \frac{b+a}{2\sqrt{a}}$ ,

where $a$ is the closest perfect square to $b$ . For example,

$\sqrt{26} \approx \displaystyle \frac{26+25}{2\sqrt{25}} = 5.1$ .

I had not seen this trick before — at least stated in these terms — and I’m definitely not a fan of computational tricks without an explanation. In this case, the approximation is a straightforward consequence of a technique we teach in calculus. If $f(x) = (1+x)^n$ , then $f'(x) = n (1+x)^{n-1}$ , so that $f'(0) = n$ . Since $f(0) = 1$ , the equation of the tangent line to $f(x)$ at $x = 0$ is

$L(x) = f(0) + f'(0) \cdot (x-0) = 1 + nx$ .

The key observation is that, for $x \approx 0$ , the graph of $L(x)$ will be very close indeed to the graph of $f(x)$ . In Calculus I, this is sometimes called the linearization of $f$ at $x =a$ . In Calculus II, we observe that these are the first two terms in the Taylor series expansion of $f$ about $x = a$ .

For the problem at hand, if $n = 1/2$ , then

$\sqrt{1+x} \approx 1 + \displaystyle \frac{x}{2}$

if $x$ is close to zero. Therefore, if $a$ is a perfect square close to $b$ so that the relative difference $(b-a)/a$ is small, then

$\sqrt{b} = \sqrt{a + b - a}$

$= \sqrt{a} \sqrt{1 + \displaystyle \frac{b-a}{a}}$

$\approx \sqrt{a} \displaystyle \left(1 + \frac{b-a}{2a} \right)$

$= \sqrt{a} \displaystyle \left( \frac{2a + b-a}{2a} \right)$

$= \sqrt{a} \displaystyle \left( \frac{b+a}{2a} \right)$

$= \displaystyle \frac{b+a}{2\sqrt{a}}$ .

One more thought: All of the above might be a bit much to swallow for a talented but young student who has not yet learned calculus. So here’s another heuristic explanation that does not require calculus: if $a \approx b$ , then the geometric mean $\sqrt{ab}$ will be approximately equal to the arithmetic mean $(a+b)/2$ . That is,

$\sqrt{ab} \approx \displaystyle \frac{a+b}{2}$ ,

so that

$\sqrt{b} \approx \displaystyle \frac{a+b}{2\sqrt{a}}$ .

Why Should Physicists Study History?

I’ve always enjoyed reading about the history of both mathematics and physics, and so I really appreciated this perspective from Physics Today magazine about the importance of this field. One of many insightful paragraphs:

And a more human physics is a good thing. For starters, it makes physics more accessible, particularly for students. Many promising students drop out of the sciences because the material seems disembodied and disconnected from their lives. Science education researchers have found that those lost students “hungered—all of them—for information about how the various methods they were learning had come to be, why physicists and chemists understand nature the way they do, and what were the connections between what they were learning and the larger world.” Students can potentially lose the wonder and curiosity that drew them to science in the first place. Historical narratives naturally raise conceptual, philosophical, political, ethical, or social questions that show the importance of physics for the students’ own lives. A field in which people are acknowledged as people is much more appealing than one in which they are just calculating machines.

The whole article can be found here: https://physicstoday.scitation.org/doi/full/10.1063/PT.3.3235

Texas slide rule competitions

I got a kick out of reading this retrospective of Texas high school slide rule competitions… including a 1959 picture of Janis Joplin on her high school slide rule team and a 1980 Dallas Morning News article eulogizing the competition.

https://mikeyancey.com/uil-slide-rule-resources/

Adventures in Fine Hall: Princeton mathematics in the 1930s

I enjoyed reading this retrospective about the famous mathematicians at Princeton in the 1930s: https://paw.princeton.edu/article/adventures-fine-hall

From the opening two paragraphs:

The year was 1933. Members of the University’s mathematics department and the Institute for Advanced Study were celebrating the Institute’s opening with a party at the Princeton Inn, which is now Forbes College. “By chance,” an attendee later recalled, he entered just behind the Institute’s most famous faculty member, Albert Einstein. “As we walked across the lobby of the hotel, a Princetonian lady, of the Princetonian variety, strolled toward us. She was fairly tall and almost as wide, beautifully dressed, and she had an air of dignity. She strolled up to Einstein, reached out, put her hand up on Einstein’s head, ruffled his hair all over the place, and said, ‘I have always wanted to do that.’ ”

The source of this marvelous anecdote is Edward McShane, a distinguished mathematician, and the context is an intriguing series of interviews that the University conducted in the 1980s with people who had studied in the mathematics department in the 1930s. These interviews sought to capture the spirit of mathematics at Princeton during a golden age, a time when Einstein, Kurt Gödel, John von Neumann, and other analytical greats crossed paths on campus. In the process, the interviews captured something unexpected: a catalog of weirdness, a palette of colorful and off-kilter adventures that were going on in the background while the big papers were being written.

Hamilton Day

No, not that Hamilton.

Courtesy of Slate magazine and mathematics journalist Katharine Merow: Today is the anniversary of the great insight that led William Rowan Hamilton to the discovery of quaternions. Details can be found here: http://www.slate.com/articles/health_and_science/science/2016/10/we_should_celebrate_hamilton_day_a_mathematical_holiday_on_oct_16.html

Or the day can be celebrated in song:

Euler and 1,000,009

Here’s a tale of one the great mathematicians of all time that I heard for the first time this year: the great mathematician published a mistake… which, when it occurs today, is highly professionally embarrassing to modern mathematicians. From Mathematics in Ancient Greece:

In a paper published in the year 1774, [Leonhard] Euler listed [1,000,009] as prime. In a subsequent paper Euler corrected his error and gave the prime factors of the integer, adding that one time he had been under the impression that the integer in question admitted of the unique partition

$1,000,009 = 1000^2 + 3^2$

but that he had since discovered a second partition, namely

$1,000,009 = 235^2 + 972^2$ ,

which revealed the composite character of the number.

See Wikipedia and/or Mathworld for the details of how this allowed Euler to factor $1,000,009$ .

Factoring Mersenne “primes”

I love hearing and telling tales of legendary mathematicians. Today’s tale comes from Frank Nelson Cole and definitely comes from the era before calculators or computers. From Wikipedia:

On October 31, 1903, Cole famously made a presentation to a meeting of the American Mathematical Society where he identified the factors of the Mersenne number 2⁶⁷ − 1, or M₆₇. Édouard Lucas had demonstrated in 1876 that M₆₇ must have factors (i.e., is not prime), but he was unable to determine what those factors were. During Cole’s so-called “lecture”, he approached the chalkboard and in complete silence proceeded to calculate the value of M₆₇, with the result being 147,573,952,589,676,412,927. Cole then moved to the other side of the board and wrote 193,707,721 × 761,838,257,287, and worked through the tedious calculations by hand. Upon completing the multiplication and demonstrating that the result equaled M₆₇, Cole returned to his seat, not having uttered a word during the hour-long presentation. His audience greeted the presentation with a standing ovation. Cole later admitted that finding the factors had taken “three years of Sundays.”

The Map of Mathematics

This video made the rounds earlier this year:

My Favorite One-Liners: Part 52

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s story is a continuation of yesterday’s post.

When I teach regression, I typically use this example to illustrate the regression effect:

Suppose that the heights of fathers and their adult sons both have mean 69 inches and standard deviation 3 inches. Suppose also that the correlation between the heights of the fathers and sons is 0.5. Predict the height of a son whose father is 63 inches tall. Repeat if the father is 78 inches tall.

Using the formula for the regression line

$y = \overline{y} + r \displaystyle \frac{s_y}{s_x} (x - \overline{x})$ ,

we obtain the equation

$y = 69 + 0.5(x-69) = 0.5x + 34.5$ ,

so that the predicted height of the son is 66 inches if the father is 63 inches tall. However, the prediction would be 73.5 inches if the father is 76 inches tall. As expected, tall fathers tend to have tall sons, and short fathers tend to have short sons. Then, I’ll tell my class:

However, to the psychological comfort of us short people, tall fathers tend to have sons who are not quite as tall, and short fathers tend to have sons who are not quite as short.

This was first observed by Francis Galton (see the Wikipedia article for more details), a particularly brilliant but aristocratic (read: snobbish) mathematician who had high hopes for breeding a race of super-tall people with the proper use of genetics, only to discover that the laws of statistics naturally prevented this from occurring. Defeated, he called this phenomenon “regression toward the mean,” and so we’re stuck with called fitting data to a straight line “regression” to this day.