Tag Archives: Math

Counting your chickens before numbers

6000-2200 bc. Eventually people figured out it was a lot easier to raise livestock—animals that provide food—than chasing after them with bows and arrows. Human beings domesticated certain kinds of animals, like poultry (chickens), cattle, sheep and goats.

If you wanted to tell somebody how many chickens you owned, you couldn’t, because there weren’t any symbols for numbers.

My version of a clay chicken token.

It was pretty important to know how many chickens—or goats, or sheep—you owned. Sometimes people would keep a bag of pebbles. Each pebble represented a chicken or a goat. Eventually someone had the idea to make little clay chickens and goats. At the end of every day, the animals went back into their pens. As each chicken entered the coop, you could keep track by putting a clay chicken in your bag for every real chicken. As each goat entered the pen, you put a clay goat in your bag.

You get what’s happening here? We switched from making images of animals as grand wall paintings to inventing a token or symbol (a clay chicken) that represents a unit (a real chicken). That’s a big deal. These symbols were the first step toward a written language.

“What do you mean by that, Manders? Stop spewing gibberish!” Okay, okay. To make my point more clearly: look at the cartoons in this post. A cartoon will only represent what it was drawn to represent. Now look at the letters in this text. They’re symbols. The letters can be rearranged to make different words, to say anything you like. It will take a loooooong time to get from chicken tokens to an alphabet, but we’re on our way.


Early Counting Systems

View at Medium.com

Back to the beginning of The Western Civ User’s Guide to Reading & Writing.

East is east and west is west

Navigators still faced the problem of not knowing how far east or west they were. Latitude is how far north or south you are. You can tell that with an astrolabe. Longitude is how far east or west.
It was a problem Amerigo Vespucci tried to solve. In 1502, he wrote: “…I learned [my longitude] … by the eclipses and conjunctions of the Moon with the planets…” He was trying to find longitude by observing the Moon’s and Mars’ positions in relation to the Earth. Not only was this an overly-complicated method, it had several drawbacks—mainly it only worked during a specific astronomical event.


I’d be a clod not to link Rudyard Kipling’s poem, which I quoted in the title above—http://www.kiplingsociety.co.uk/poems_eastwest.htm

While we’re at it, here’s Bob Hope and Jane Russell in The Paleface, singing Buttons & Bows.

Who doesn’t like π?

Pi, or π, is a letter from the Greek alphabet used by mathematicians. π signifies this weird number: 3.14159265359… or 3.14 for short. It is the ratio of a circle’s circumference to its diameter. Circumference is how big around a circle is. The diameter is how wide a circle is from side to side if you draw a line through its center. Diameter x 3.14 = circumference. This is true of any circle, no matter how big or small.

MrNystrom has a great video about how to think about π.

Pi has been around for 4000 years, but Archimedes of Syracuse (287–212 bc) was the first mathematician to calculate π accurately.


Archimedes came up with his best ideas in the bath tub.

So back to the Romans. If you want to measure miles across the Roman Empire, how would you do it? Counting steps and paces as you march along isn’t very accurate—it’s too easy to lose count. What if you used a circle—like a wheel? I mentioned that chariot wheels were made a standard size, like many things in the Roman Empire.


A chariot wheel is 4 feet across at its widest point—that’s the diameter. Let’s calculate a chariot wheel’s circumference—how big around the wheel is. You can calculate the circumference by multiplying its diameter by π. π = 3.14. Four feet x 3.14 = 12.56 feet.

Now, there are 5280 feet in one mile. 5280 divided by 12.56 = 42 revolutions of the chariot wheel. All we have to do now is count every time the wheel goes around. At 42 times, we’ll know we’ve reached a mile.

Counting how many times a chariot wheel goes around still seems like a big pain in the neck, doesn’t it? Archimedes thought so, too.

Side note: March 14, 3/14, is known as π Day. On π Day a grocery store in my town sells pies for $3.14.

pie day559

Back to the beginning of The Western Civ User’s Guide to Time & Space

Base Sixty

Today we count using Base Ten (1-2-3-4-5-6-7-8-9-10, then 11-12-13-14-15-16-17-18-19-20, 21 to 30, 31 to 40, 41 to 50 and so on). Here’s something really interesting about the Sumerians. They counted numbers using Base Sixty! Fractions hadn’t been invented yet, so 60 was actually a handy base for counting. Sixty can be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Try it!

Base Sixty is natural, it occurs in the world around us. By watching the sky, the Sumerians saw the Moon make a complete cycle 12 times a year. Each cycle (month) takes about 30 days (one day = the time it takes Earth to completely revolve around her axis). That’s about 360 days per year (one year = the time it takes Earth to travel around the Sun). These numbers fit neatly into the Base Sixty way of counting.

Do we still use Base Sixty? You betcha! A ruler is 12 inches long—12 x 5 = 60. We buy eggs and doughnuts by the dozen. Can you think of any other examples?

How about a clock? There are 12 hours on the face of an analogue clock—12 x 5 = 60. Sixty minutes in an hour; sixty seconds in a minute. What about 24 hours in the day—does that work? Nope, sixty doesn’t evenly divide by 24. But a protractor uses Base Sixty—a circle is divided into 360 degrees. If you were to mark 24 hours around a protractor, each hour would use 15 degrees (hang onto that thought—it will become important later on in the show).

I bought this one at Protractor Supply.

I can hear you saying, “Hold on, Manders. Years aren’t 360 days long. They’re 365 days and six hours!” Okay, I have to admit, that’s a good point. Sumerian astronomers were fantastic at math and did mostly everything right—but they got one really important piece of information wrong. They thought the Earth was the center of the universe with the Sun, Moon and planets revolving around her. It’s impossible to make the lunar year (12 cycles of the Moon) agree exactly with the solar year (Earth’s trip around the Sun), so we had to add 5 extra days, plus one day every fourth year, to make it work on a calendar. That’s not really the best solution, as we will see.