In the 1962 movie, *Lawrence of Arabia*, there is a scene where Auda abu Tayi (Anthony Quinn), the leader of the powerful local Howeitat tribe, declares in exasperation, “I must find something honorable.” And so it was that, with lies reaching an enormous scale – a blizzard of lies – Ersjdamoo determined upon finding something honorable and found it in the number “e”. (Background: Euler’s Identity: “God Equation” or “Cthulhu Equation”?, Ersjdamoo’s Blog, April 19, 2015.)

Further now in my quest for the mysteries of “e”, a panorama of truths continues to unfold. Readers of this blog have already seen how the five most common constants of mathematics: e, π (Pi), i (imaginary number constant), 0, and 1 all conspire in a special case of the Euler Identity. (Flash! “Euler” is pronounced “Oiler”.) Now something new has unveiled itself from the infinity of “e”: an eerie logarithmic connection.

“There is a joke in mathematics publishing that for every equation in a book, half the readers go away.” [1] With this in mind, I will keep the equations here to a minimum. Complete exposition of the logarithmic connection to “e” can be found in Chapter 12 of Morris Kline’s book, *Calculus: An Intuitive and Physical Approach*. (Mineola, NY: Dover Publications, 1998).

Kline seeks to find the derivative of y = log x. The first thing to note is that this is not a “natural log”, a log already having a base of “e”. Information coming my way has it that logarithms of base 10 are written in the form y = log x, and that logarithms of base “e” (“natural logs”) are written in the form y = ln x.

Again, avoiding the complete exposition found in Kline’s book (so as not to frighten away casual readers with equations), the gist is that Kline arrives through derivation of y = log x at the following: dy/dx = 1/x_{0} log(1 + dx/x_{0})^{x0/dx}.

Well that certainly is a frightening statement! But to make a long story short, Kline takes from this the simpler expression: (1 + t)^{1/t}. And that is where I noticed a startling similarity to the origin of “e”! For “e” is the limit as we approach infinity of (1 + 1/n)^{n}.

Compare the two: (1 + t)^{1/t} and (1 + 1/n)^{n}.

Kline next takes his (1 + t)^{1/t} and plugs in various numbers – this time approaching zero instead of approaching infinity. And what does he find? As t approaches zero, (1 + t)^{1/t} approaches “e”! (e = 2.71828…)

In a world of lies, this is great stuff (for me, anyway). Wearied by the barrage of lies (called “news”), I had desperately grasped for something – anything – true. For me, this meant dusting off the old math books and getting a few new ones. For others, the quest for “something honorable” (something true) might lead elsewhere, perhaps to the Bible or the Koran. I had been influenced in my youth by René Descartes and his book, *Meditations on First Philosophy*. In middle age, Descartes had also despaired that truth could be found. The senses were “the great deceivers”; what did he really know for sure? After discarding all sensory input (“the great deceivers”), one thing began to emerge as still rock-solid: 2 + 2 = 4, regardless of whether or not his senses lied. And so it was that, drowning in a sea of lies, I almost instinctively reached out to mathematics to keep me afloat in my hour of peril.

And what about Auda abu Tayi? “I must find something honorable,” he had cried out. Did Auda abu Tayi at last find it? In the movie *Lawrence of Arabia*, Auda abu Tayi did indeed, in the form of a magnificent horse.

——- Sources ——-

[1] “Not exactly war of all againts [sic] all, but still a very damaging fight”, by Mariano Apuya Jr, April 9, 2014. Amazon reader review of *Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World*, by Amir Alexander. Scientific American / Farrar, Straus and Giroux; April 8, 2014.