Early Timekeeping

Why are there sixty seconds in a minute, sixty minutes in an hour, yet twenty-four hours in a day? The answer is because modern timekeeping derives from the base-60 number system.

It is believed that Sumerians of Mesopotamia used their phalanges to count. They counted increments of 12 with one hand's four fingers, each of which has three bones, and tracked them with the other hand's five fingers: 12, 24, 36, 48, 60.

Early civilization calendars were often lunisolar, based on the phases of the moon—roughly aware of the sun's yearly 365-day orbit. Though they were somewhat imprecise, many resembled the 12-month calendar we know today. Ancient calendars, however, would often have extra days or months periodically added for alignment purposes.

The Sumerians had no tradition for referring to the length of time we call a "week," nor did they identify months. They simply observed months and years. Later, the Babylonians would put forth the notion of the "week," as well as move to use a solar, rather than lunisolar, calendar.

A history of calendars: https://en.wikipedia.org/wiki/History_of_calendars

Currently Reading: Ben Franklin's Autobiography

Today, while updating Windows virtual machines for Patch Tuesday, I found myself re-reading Ben Franklin's autobiography.

Two fun excerpts:

At my first admission into this printing-house I took to working at press, imagining I felt a want of the bodily exercise I had been us'd to in America, where presswork is mix'd with composing. I drank only water; the other workmen, near fifty in number, were great guzzlers of beer. On occasion, I carried up and down stairs a large form of types in each hand, when others carried but one in both hands. They wondered to see, from this and several instances, that the Water-American, as they called me, was stronger than themselves, who drank strong beer! We had an alehouse boy who attended always in the house to supply the workmen. My companion at the press drank every day a pint before breakfast, a pint at breakfast with his bread and cheese, a pint between breakfast and dinner, a pint at dinner, a pint in the afternoon about six o'clock, and another when he had done his day's work. I thought it a detestable custom; but it was necessary, he suppos'd, to drink strong beer, that he might be strong to labour. I endeavoured to convince him that the bodily strength afforded by beer could only be in proportion to the grain or flour of the barley dissolved in the water of which it was made; that there was more flour in a pennyworth of bread; and therefore, if he would eat that with a pint of water, it would give him more strength than a quart of beer. He drank on, however, and had four or five shillings to pay out of his wages every Saturday night for that muddling liquor; an expense I was free from. And thus these poor devils keep themselves always under.

In the excerpt below, Ben Franklin recounts teaching others at the printing house how to swim—as well as leaping into the river in front of visitors at the college and coffee shop, surprising onlookers:

At Watts's printing-house I contracted an acquaintance with an ingenious young man, one Wygate, who, having wealthy relations, had been better educated than most printers; was a tolerable Latinist, spoke French, and lov'd reading. I taught him and a friend of his to swim at twice going into the river, and they soon became good swimmers. They introduc'd me to some gentlemen from the country, who went to Chelsea by water to see the College and Don Saltero's curiosities. In our return, at the request of the company, whose curiosity Wygate had excited, I stripped and leaped into the river, and swam from near Chelsea to Blackfriar's, performing on the way many feats of activity, both upon and under water, that surpris'd and pleas'd those to whom they were novelties.

Binary, IPv4, and Subnets

The IPv4 protocol which we broadly (but not totally) use today rests on an addressing system that was designed in the 1970s and formally published in 1980.

James Garfield's Pythagorean Proof

Today I learned James Garfield, who once worked as a lawyer, Civil War General, and served as the 20th President of the United States, was math savvy and published a novel Pythagorean theorem proof.^[1]

\[ \text{Area}_{\text{trapezoid } ACED} = \frac{1}{2} \cdot (AC + DE) \cdot CE = \frac{1}{2} \cdot (a + b) \cdot (a + b) = \frac{(a + b)^2}{2} \] \[ \begin{aligned} \text{Area}_{\text{trapezoid } ACED} &= \text{Area}_{\Delta ACB} + \text{Area}_{\Delta ABD} + \text{Area}_{\Delta BDE} \\ &= \frac{1}{2}(a \times b) + \frac{1}{2}(c \times c) + \frac{1}{2}(a \times b) \end{aligned} \] \[ (a + b) \times \frac{1}{2}(a + b) = \frac{1}{2}(a \times b) + \frac{1}{2}(c \times c) + \frac{1}{2}(a \times b) \] \[ a^{2} + b^{2} = c^{2} \]

Small Pieces

We can take this in smaller pieces. First, we can find the area of the right-angled trapezoid with the following equation:

\[ \text{Area}_{\text{trapezoid}} = \frac{1}{2} \cdot (a + b) \cdot (a + b) = \frac{(a + b)^2}{2} \]

We can find the area of each of the two outer triangles with the following:

\[ \text{Area}_{\text{triangle}} = \frac{ab}{2} \]

And the area of the inner triangle with:

\[ \text{Area}_{\text{inner triangle}} = \frac{c^2}{2} \]

Proof

Reducing, we can go to the end, beginning with our substituted and now simplified area equation demonstrated above:

\[ \frac{(a + b)^2}{2} = 2 \cdot \frac{ab}{2} + \frac{c^2}{2} \]

Then we expand \( (a + b)^2 \) on the left hand side. And our equation on the right can also be simplified since we're both multiplying and dividing \( ab \) by 2:

\[ \frac{a^2 + 2ab + b^2}{2} = ab + \frac{c^2}{2} \]

Multiply both sides by 2 to eliminate denominators:

\[ a^2 + 2ab + b^2 = 2ab + c^2 \]

Lastly, subtract \( 2ab \) from both sides:

\[ a^2 + b^2 = c^2 \]

---

Footnotes

1. Mathematical Treasure: James A. Garfield’s Proof of the Pythagorean Theorem ↩︎

Euler's Equation

Euler's number, e, the constant 2.71828, is the base of the natural logarithms. Given n approaching infinity, Euler's number is the limit of:

\begin{align*}\displaystyle{\displaylines{(1 + 1/n)n}}\end{align*}

It's used frequently abroad across the sciences. It can also be elegantly expressed as an infinite series, like so:

\begin{align*} {\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .} \end{align*}

Separately, the imaginary unit i, \({\displaystyle {\sqrt {-i}}}\), represents the imaginary solution to the quadratic equation, x² + 1 = 0. The value can also be used to extend real numbers to complex numbers.

And π is pi, the irrational number we all know and love, roughly approximate to 3.14159, representing the ratio of the circle's circumference to its diameter.

While it isn't absolutely understood, we can join the three numbers in a seemingly bizarre proof that just works.

\( {\displaystyle e^{i\pi }=-1} \)