Postscript of a letter from Benjamin Franklin to the Abbé André Morellet, July 1779:

P.S. To confirm still more your piety and gratitude to Divine Providence, reflect upon the situation which it has given to the elbow. You see in animals, who are intended to drink the waters that flow upon the earth, that if they have long legs, they have also a long neck, so that they can get at their drink without kneeling down. But man, who was destined to drink wine, is framed in a manner that he may raise the glass to his mouth. If the elbow had been placed nearer the hand, the part in advance would have been too short to bring the glass up to the mouth; and if it had been nearer the shoulder, that part would have been so long that when it attempted to carry the wine to the mouth it would have overshot the mark, and gone beyond the head; thus, either way, we should have been in the case of Tantalus. But from the actual situation of the elbow, we are enabled to drink at our ease, the glass going directly to the mouth.

“Let us, then, with glass in hand, adore this benevolent wisdom; — let us adore and drink!”

This just caught my eye in an old issue of the Mathematical Gazette, a note from P.G. Wood. Suppose we’re designing a cylinder that’s closed at both ends and must encompass a given volume. What relative dimensions should we give it in order to minimize its surface area?

A young student thought, well, if we slice the cylinder with a plane that passes through its axis, the plane’s intersection with the cylinder will form a rectangle. And if we spin that rectangle, it’ll sweep out the surface area of the cylinder. So really we’re just asking: Among all rectangles of the same area, which has the smallest perimeter? A square. So the cylinder’s height should equal its diameter.

It turns out that’s right, but the student had overlooked something. The fact that the volume of the cylinder is fixed doesn’t imply that the area of the rectangle is fixed. We don’t know that.

Wood wrote, “We seem to have arrived at the right answer by rather dubious means.”

(P.G. Wood, “73.5 Interesting Coincidences?”, Mathematical Gazette 73:463 [1989], 33-33.)

Strategies to memorize π sometimes rely on devising a sentence with words of representative lengths. Isaac Asimov offered this one:

How I want a drink, alcoholic, of course, after the heavy lectures involving quantum mechanics!

Count the letters in each word and you get 3.14159265358979. That’s not the best strategy, though: What shall we do about zero? And it’s hard to reel off the digits impressively for friends when you have to stop and count the letters in each word.

Arthur Benjamin, a mathematician at Harvey Mudd College, suggests using a phonetic code instead:

1 = t or th or d
2 = n
3 = m
4 = r
5 = l
6 = sh, ch, or j
7 = k or hard g
8 = f or v
9 = p or b
0 = s or z

Once we’ve memorized this list, we can turn a whole string of digits into a single word simply by inserting vowels between the consonants. So, for example, 314 could be represented by meter, motor, meteor, matter, mother, etc. The consonants h, w, and y, which don’t appear in the list, can be used like additional vowels.

Using this method, it takes only three sentences to encode the first 60 digits of π:

3. 1415 926 5 3 58 97 9 3 2 384 6264
My turtle Pancho will, my love, pick up my new mover Ginger.

3 38 327 950 2 8841 971
My movie monkey plays in a favorite bucket.

69 3 99 375 1 05820 97494
Ship my puppy Michael to Sullivan’s backrubber!

Now we just need a way to remember the sentences …

(Arthur T. Benjamin, “A Better Way to Memorize Pi: The Phonetic Code,” Math Horizons 7:3 [February 2000], 17.)

The invention of logarithms came on the world as a bolt from the blue. No previous work had led up to it, foreshadowed it, or heralded its arrival. It stands isolated, breaking in upon human thought abruptly without borrowing from the work of other intellects or following known lines of mathematical thought.

— Lord Moulton on the 300th anniversary of John Napier’s 1614 book Description of the Wonderful Canon of Logarithms. E.W. Hobson called the invention “one of the very greatest scientific discoveries that the world has seen.”

Usain Bolt is such a great sprinter that his distinctions may extend to other worlds.

In 2013, University of Leicester physics undergraduate Hannah Lerman and her colleagues determined that the Jamaican athlete was one of the few humans who could get aloft on Saturn’s moon Titan with wings strapped to his arms.

Factoring in Titan’s gravity and atmospheric density, Lerman found that a person could take flight in a normal-sized wingsuit only if they could run at 11 meters per second.

“This speed has been reached but only by the fastest human runners, for example, Usain Bolt, who ran almost 12 m/s,” Lerman wrote. “For an average human to take off with the standard wingsuit they would require some sort of propulsion device to give them enough speed to take off.”

(H. Lerman, B. Irwin, and P. Hicks, “P5_1 You Can Fly,” Journal of Physics Special Topics, University of Leicester, Oct. 22, 2013.)

A paradox by the German mathematician Martin Löb:

Let A be any sentence. Let B be the sentence: ‘If this sentence is true, then A.’ Then a contradiction arises.

Here’s the contradiction. B makes the assertion “If B is true, then A.” Now consider this argument. Assume B is true. Then, by B, since B is true, A is true. This argument shows that, if B is true, then A. But that’s exactly what B had asserted! So B is true. And therefore, by B, since B is true, A is true. And thus every sentence is true, which is impossible.

(Lan Wen, “Semantic Paradoxes as Equations,” Mathematical Intelligencer 23:1 [December 2001], 43-48.)

In the January 2001 issue of the College Mathematics Journal, University of Calgary mathematician Jonathan Schaer offers this simple proof that arctan 1 + arctan 2 + arctan 3 = π. The sum of the angles of this large triangle is 180°. And the diagram shows that its lower left angle is arctan 3, its lower right angle is arctan 2, and its top angle is part of an isosceles right triangle. So arctan 1 + arctan 2 + arctan 3 = 180°, or π radians. “No words, even no symbols!”

(“Miscellanea,” College Mathematics Journal 32:1, 68-71.)

Welsh mathematician Robert Recorde’s 1543 textbook Arithmetic: or, The Ground of Arts contains a nifty algorithm for multiplying two digits, a and b, each of which is in the range 5 to 9. First find (10 – a) × (10 – b), and then add to it 10 times the last digit of a + b. For example, 6 × 8 is (4 × 2) + (10 × 4) = 48.

This works because (10 – a)(10 – b) + 10(a + b) = 100 + ab, and it saves the student from having to learn the scary outer reaches of the multiplication table — they only have to know how to multiply digits up to 5.

(From Stanford’s Vaughan Pratt, in Ed Barbeau’s column “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 38:1 [January 2007], 43-46.)