Tag Archives: Western Civilization

Archimedes and his odometer

Archimedes was a Greek mathematician who specialized in measuring space. He was certain that there must be a way to accurately measure how much space is in a circle—area—or how much space is in a sphere or a cylinder—volume (he figured out how to measure volume when he noticed that a certain amount of water spilled out when he got into a bath tub). Archimedes was influenced by other great mathematicians, like Pythagorus and Euclid.

Archimedes invented many wonderful machines, like a screw for drawing up water, or catapults that were used to fight off invading navies. Although we don’t have his plans for it, Archimedes is said to have invented a way to measure distance. This machine is called an odometer.

Archimedes’ odometer operated on the idea that every time a wheel goes around, it travels its own circumference. The odometer adds up those circumferences and marks when the wheel has traveled a mile. In our last post, we showed how a standard Roman chariot wheel goes around 42 times to travel a mile.

We know about Archimedes’ odometer because the Roman military engineer Marcus Vitruvius Pollio (80–70 bc – 15 bc), or Vitruvius for short, wrote about it in his 10-volume book De Architectura. Engineers build stuff. As the Roman Empire expanded, the army took along a corps of engineers to build fortifications; siege engines; bridges; tunnels; aqueducts to provide water; and roads. These engineers did such a good job that you can still find Roman bridges, aqueducts and roads today.

Emperor Caesar Augustus wanted to know exactly how big the empire was and decreed that mile markers should be put up along the newly-built roads. Vitruvius decided to build Archimedes’ odometer to accurately measure the miles.

We only know what Vitruvius’ odometer looked like from a fanciful drawing. We don’t know exactly how it worked. Some people, including Leonardo da Vinci, have come up with some pretty good guesses about how it worked. You can see Leonardo’s drawings here—plus, you can even download plans if you’d like to build one yourself! Now that’s cool.

We do know that every time the chariot wheel goes completely around, it moves other gears. The other gears are set up to mark a mile at the 42nd revolution of the chariot wheel. The trick is gear ratio—meaning some gears are bigger, some gears have more teeth. If the gear on the drive shaft has only one tooth and the gear holding the marbles has 42, the marble-gear moves 1/42 of a revolution every time the chariot wheel goes completely around. At the 42nd revolution, a hole with a marble lines up with a hole underneath the gear and the marble drops into a bucket. Each dropped marble represents one mile traveled.



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Measuring long distances

The Romans were geniuses at organization. The way they organized their government and army/navy is how they could maintain such a huge empire. The Romans found that standardization really helped—from soldiers’ armor to the width of a chariot wheel to money to constructing roads.

Good roads are important to a maintain a huge empire, especially for moving armies quickly from one place to another. It’s vital to know exact distances, too, if you are planning a big march with thousands of soldiers. We know that one Roman mile is 1,000 passus—paces—or 5,000 gradus—steps. The problem is, it’s too easy to lose count of all those steps. How do you measure a mile and know for sure you didn’t lose count?

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The future!

Can you see into the future? Thinking in linear time allows us to think about the future. It’s not easy, of course. People do use a timeline to plan projects and life goals, so that they can make them happen. In fact, I’m using a timeline for this project—writing this history which I plan to make into a book. Do you think about what you’ll be doing in 5 years? Successful people plan for the future. Holy cow, I sound like I’m selling life insurance.

The Torah (what Christians call the Old Testament), with its timeline of many generations of Jews, was read by ordinary people and changed the way ordinary people think about time and themselves.

In the Torah, ordinary people became important. Religious stories of other cultures were about gods and goddesses. Mortal beings had supporting roles in those stories, like when the mortal Paris judged a goddess beauty contest, or the gods created Enkidu to hang out with the semi-divine Gilgamesh. You don’t get a sense that the gods and mortals have a destiny together, because the stories don’t talk about a future. On the other hand, the stories in the Torah are about ordinary mortals who share a past and future with one God. When ordinary mortals—everybody, us—see ourselves as part of a destiny, the way we think about ourselves is changed. A person with a future, a destiny, has free-will and the mindset to break free of a cycle. You can see how a person with the inheritance of generations of history—aligned with God’s—has more free-will than the person who lives only in a cycle, the now, the present.


Thinking in linear time changes our brains. Do you like history? History lovers keep a timeline in their heads. My dad loves history. He was a handy guy to have around when I had history homework. You can give my dad a date in history and he’ll tell you all about what was going on in the world at the time, what people were likely wearing, what they ate, how they got around, who were the big political players. He taught me how to research, how to use a library. When I was awarded the contract to illustrate A Prairie Dog for the President, I phoned Pop who started giving me facts about Lewis & Clark—just off the top of his head. Don’t watch a historical movie with my dad—if they monkeyed with the research he’ll take it apart!

A timeline is a foundation you can build on. If you have a general idea where the big events are on a timeline, you can add new eras and dates in their proper places. A timeline is a fantastic tool for organizing the way you think.

Happy Father’s day!

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How he did it

Just to recap from the last couple of posts: First, Eratosthenes guessed the Earth was round like a ball. Second, he knew that at noon on the Summer solstice, the Sun shone directly overhead in the town of Syene, which was 5,000 stadia south of Eratosthenes.

Third, Eratosthenes guessed the Sun was really big—huge, even. He noticed that shadows cast by the Sun are all parallel, so all the Sun’s rays must be parallel, too. Parallel means 2 or more lines that never touch—they stay the same distance from each other. Think train tracks.

SO, Eratosthenes—in Alexandria, 5,000 stadia north of Syene—put a stick (like the gnomon of a sundial) in the ground and made sure it was plumb. That stick was pointing down to the center of the Earth. If Earth is round, the well in Syene and Eratosthenes’ stick won’t be parallel, right? They’ll be at an angle to each other. Eratosthenes didn’t know exactly how many degrees that angle was, but in Alexandria at noon on June 21st, his stick cast a shadow.

There weren’t any shadows in Syene, because the Sun was directly overhead. He measured the stick, he measured the shadow, and used those measurements to draw an angle. The angle turned out to be a little over 7 degrees, or 1/50 of a circle.

Eratosthenes knew the distance from Alexandria to Syene was 5,000 stadia. He multiplied 5,000 by 50 to get the circumference of the Earth—250,000 stadia. In modern measurements that works out to be 28,738.418 miles or 46,250 kilometers.

The actual polar circumference of Earth is about 24,860 miles or just a bit over 40 thousand kilometers. The stadion Eratosthenes used may have been a little different from the standard unit. But even today, right now, if you search the internet for the circumference of the Earth, you won’t get just one answer.

Eratosthenes was a genius who used what he knew and observed, along with what he guessed at, to calculate something that no one knew—and he did it pretty accurately.

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A plumb line or plumb bob.

Last post I rattled on about how the well at Syene was plumb.

‘Plumb’ means straight up-and-down. Builders used to use a plumb line—a string with a weight tied onto it—to check that their walls were straight up-and-down. A plumb line will always point to the center of the Earth. Nowadays they use a spirit level. Builders who dug wells made sure that the hole they dug was plumb—pointing directly toward the Earth’s center.

Why was it so important to Eratosthenes that the well at Syene (a town located on the Tropic of Cancer) was pointing directly at the Earth’s center at noon on June 21st—the Summer solstice?

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Head-Librarian Eratosthenes explains to a student how to laminate a dust jacket.

In an earlier post, I hinted about how over 2,000 years ago somebody calculated the circumference (how big around) of the Earth. This guy did it using only a well, a protractor and a stick (okay, maybe instead of a stick he used a column, but you could use a stick and get the same result).

I’m talking about Eratosthenes, the head librarian at the Great Library of Alexandria. Alexandria is an ancient city in Egypt, located where the Nile River flows into the Mediterranean Sea. Many scholars lived in Alexandria—like Ctesibius, who perfected the water-clock.

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Summer solstice

Okay, here comes some sciency stuff.

Summer solstice happens twice a year—once in the Northern Hemisphere (June 21) and once in the Southern Hemisphere (December 22). Those are the two days when the Sun spends the most time in the sky.

If you have a globe handy, you’ll see that there’s a line that goes all the way around the middle of the Earth. That’s the equator. It separates the Earth into a northern half and a southern half. Each half is called a hemisphere—’hemi’ means ‘half’ and ‘sphere’ is another word for ‘globe’.

The Sun appears to travel across the sky, but ackshually the Earth is traveling around the Sun and spinning on her axis at the same time. An axis is the line that goes through the center of the Earth from the North Pole to the South Pole. Did you notice that your globe’s axis is tilted? It’s tilted at a little over 27 degrees, because Earth is tilted at a little over 27° in relation to its orbit around the Sun. Because of this tilt, half the year (the time it takes to orbit the Sun) the Northern Hemisphere receives more of the Sun’s rays. The other half of the year, the Southern Hemisphere receives more of the Sun’s rays. The equator gets the Sun’s rays all year round, so it’s always hot there.

It takes one year for Earth to orbit the Sun. When the Northern Hemisphere is tilted toward the Sun it’s Summer there. When the Southern Hemisphere is tilted toward the Sun it’s Summer there.

Getting back to that equator—let’s look at the globe again. There are 2 other lines that also run around the Earth. In the Northern Hemisphere, there’s the Tropic of Cancer. In the Southern Hemisphere, there’s the Tropic of Capricorn. On June 21, the Northern Hemisphere tilts toward the Sun and the Tropic of Cancer gets the Sun’s rays directly. You could draw a line from the Earth’s center through the Tropic of Cancer to the Sun. On December 22, the Southern Hemisphere tilts toward the Sun and the Tropic of Capricorn gets the Sun’s rays directly. You could draw a line from the Earth’s center through the Tropic of Capricorn to the Sun.

On June 21, the Northern Hemisphere has the most daylight of any day of the year. Six months later, on December 22, the Southern Hemisphere has the most daylight of any day of the year.

Remember, when the Northern Hemisphere has the most hours of daylight (Summer solstice) the Southern Hemisphere has the fewest hours of daylight (Winter solstice). The opposite is true at solstice six months later.

Three months after solstice is the day when the day is split equally in two, with 12 hours of daylight and 12 hours of nighttime. This is the equinox. The autumnal (Fall) equinox is September 22 in the Northern Hemisphere. March 21 is the vernal (Spring) equinox.

Why am I telling you this? Well, over two thousand years ago a Greek librarian figured out exactly how big around the Earth is using only a well and a stick—and he did it at noon during the Summer solstice.

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What’s a Grecian earn?

About 14 drachmas an hour! Har har har!

Okay, can I get a do-over? I’m not entirely happy with the drawing I did for that last post about Greek measurements. Didn’t look Greek enough, or just didn’t have pizazz or something. So I drew a new version of it, this time taking the drawings on those wonderful old Grecian urns as my inspiration. Tell me what you think.

Here’s an old Greek urn—these drawings are gorgeous. I love the light terra-cotta-colored figures on a rich black background.

Here’s the new sketch.

Herodotus of Halicarnassus

Herodotus reading his Histories to the crowd

“These are the researches of Herodotus of Halicarnassus, which he publishes, in the hope of thereby preserving from decay the remembrance of what men have done, and of preventing the great and wonderful actions of the Greeks and the Barbarians from losing their due meed of glory; and withal to put on record what were their grounds of feuds.”

While the Greeks did think of time as a cycle, there was still a desire to reach back in time to remember stories from the past. We saw how Homer kept alive the story of the siege of Troy through his epic poem, the Iliad.

Herodotus (Hay ROD oh toos) was a Greek who lived in the fourth century bc and is thought to be the first historian. He wrote The Histories mostly about the wars between the Greeks and Persians and how they got started. Herodotus was a great storyteller, but what made him a historian is that he investigated, he did research—he got his information from several sources; he visited the places where the history took place; he went to the library; he interviewed people—then he arranged the information he’d gathered to explain how and why something happened.

I have a paperback copy of The Histories. It’s not an easy read, but I like to ‘dip into’ my copy and read whatever I open to. The battle of Thermopylae is in there—that story has since been made into a graphic novel and movie, The 300—and the battles of Marathon and Salamis. Herodotus included lots of oddball side-stories and observations, which are also fun. The Persian emperors Cyrus, Darius and Xerxes are in there (Xerxes is thought to be the party-boy king Ahasuerus from the biblical Book of Esther).

History is often about the big military battles and wars. I want this history, the one you’re reading right now, to be about ideas. But I’m telling you, if the Greeks hadn’t broken the will of the mighty Persian Empire and eventually beat ‘em, there would have been no flowering of art, literature, philosophy and democracy that happened after the wars—and was the Greeks’ gift to us.

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