When you were younger, did you ever harass your parents on a road trip by repeating: "Are we there yet?" It’s easy to imagine Christopher Columbus’s crew saying the same thing! He promised them a short trip to the Indies, but it seemed like it was taking "forever" and they weren’t anywhere yet!

Columbus and his crew did not have the benefit of a GPS, so they had to come up with a method to determine relative estimates of distance. Each day he went through a routine of measuring longitude—the distance he had traveled westward—and announced a number less than his calculations would predict. He was reluctant to frighten the crew, since he had promised them a much shorter trip to the Orient.

Even if Columbus had wanted to share the real numbers, he could not have done so because he lacked the standards (circumference of the Earth) and measurement tools (a reliable clock) that would have helped him to be reasonably accurate. Measurement of distance has improved over the centuries but, like Columbus, we still have a long way to go.

As you read, you may wish to stop and try the Mini-STEM activities.

Measurement Matters - Length’s Legacy

Measuring distance, mass, and time have been important for almost as long as humans could communicate. To work together, travel together, or make plans together, humans needed standards.

According to Chinese legend, Xuanyuan Huangdi (the "Yellow Emperor" whose reign began in 2697 B.C.E.) created the first Chinese measurement units. Their length unit, the zhang, was derived from measurements of the human body. According to the Records of the Grand Historian, even the earliest Chinese citizens realized that human body units caused inconsistency. Yu the Great, another legendary figure, unified the length measurements by creating standard artifacts. Rulers with units marked on them have been unearthed in Shang Dynasty (1122 B.C.E) tombs.

In records from the Mesopotamian kingdom of Sumer and from ancient Egypt around the time of Xuanyuan Huangdi (~2700 B.C.E.), there are references to the cubit. The cubit represented the distance from a person’s elbow to the tip of their middle finger. The term cubit comes from the same root as our word "recumbent" and refers to one’s elbow. The cubit was divided into fractional cubits: the span of the hand, or the length between the tip of the little finger to the tip of the thumb (one-half cubit); the palm or width of the hand (one-sixth cubit); and the digit or width of the middle finger (one-twenty-fourth cubit). The royal cubit, which was a standard cubit enhanced by an extra palm, thus 7 palms or 28 digits long, was used in constructing buildings and monuments and in surveying in ancient Egypt.

Realizing that this length would differ from person to person, the Egyptians maintained standard artifacts, too. This is an example of one that still exists.

Mini-STEM: The Cubit

In ancient Egypt and Mesopotamia, length was measured in cubits. A cubit represented the distance from a person’s elbow to the tip of their middle finger. It was often based on the measurement of some local royalty!

• Collect data to argue that this standard could never work for team engineering.
• Measure the distance from the elbow to the tip of the middle finger to the nearest millimeter for 10 people.
• Collect the data from the entire class in a way that it can be shared, like a spreadsheet.
• Develop a histogram of the data.
• Shade the area that represents the range of Egyptian and Sumerian cubits, which is 444 - 529.2 mm.
• Calculate the variance and standard deviation of the data you collected.

Cubits enabled human teams to build together. To travel together by land, they needed a longer unit. Again, the average human became the standard—the length of a foot and the length of a pace were both important resources. The Greeks and Romans inherited the unit we call a foot from the Egyptians. The Roman foot (~296 mm) was divided into both 12 unciae (inches) (~24.7 mm) and 16 digits (~18.5 mm). For traveling, the Romans relied on a standard pace (5 feet.) A mille passus (their mile) was equivalent to 1,000 paces or double steps. Shorter soldiers just had to keep up.

The Roman mile of 5,000 feet (1,480 m) was introduced into England during the occupation. Queen Elizabeth I (who reigned from 1558 to 1603) changed the mile to 5,280 feet (~1,609 m) or 8 furlongs, a furlong being 40-rod units (~201 m) of 5.5 yards (~5.03 m) each.
 

Mini-STEM: On the March

You’ve just joined the Roman army. Your marching orders are to move 100 paces (double steps) to the north.

• Begin from the edge of a field and march 100 paces.
• Mark your endpoint and use a trundle wheel (your modern artifact) to determine the exact distance you traveled.
• Use that data to calculate the length that your personal "mile" (1,000 paces) would be.
• Calculate the mean and standard deviation of the miles that were established by the other people in your class.
• Using a map of Italy, start in Rome and determine where you could end up if you walked 20 of your "miles" each day for a month.

It was not until 1791 that the meter was proposed as a standard unit. Relying on the best measurements of the Earth at the time, the meter was set as 1/10,000,000 of the distance of the meridian from the North Pole to the Equator. (The circumference of the Earth was known to an amazingly high degree of accuracy due to the work of Eratosthenes about 300 B.C.E. He used the angles of the sun at two positions to make a calculation of about 25,000 miles. The actual value is 24,900 miles!)

The metric system was officially adopted in France in 1799, and the International Treaty of the Meter was signed by 17 nations in 1875. But those agreements still depended upon dividing what we assumed to be the length of the meridian by 10,000,000. And of course, the Earth isn’t even always the same size!
 

Measurement Matters - Mass Chaos

Travel in time to a busy marketplace in Sumeria, perhaps 4,700 years ago. You bargain for a day’s apportionment of barley in units called minas (about 0.57 kg). Travel forward in time and the mina, the shekel, and the talent became standard units of weight equivalent to known quantities of precious metals and spices. But what if you met a dishonest vendor in the marketplace? What recourse would you have to prove that his mina and that of the vendor in the next booth were different? Most would look at the equivalent volume. In Mesopotamia, a mina was considered equivalent to about .5 sila of water. The standard was a cube (called a sura) of grain. Later, a single grain was used to measure the equivalent in a precious metal or spice. The carat (which we use for diamonds) originated as the average mass of a carob seed (about .2 grams).

The Roman talent was both a standard and a coin as well. The troy pound (~373.2 g) used in England and the United States for monetary purposes, like the Roman pound, was divided into 12 ounces or unca.
 

Mini-STEM: Marketplace Mayhem

To consider how much variance there might be in a system based on barley, you can use barley itself, if you can find it, or long-grain rice.

• Use a Vernier caliper to find the smallest and largest grains in a batch.
• Calculate the mass of both.
• What is the percent difference between the largest and the smallest
• Use a dosing cup or small beaker to estimate the number of grains in 50 milliliters (1/10 of a mina).
• Use your figure for percent difference to determine how many more grains would fit in the space if the smallest grains were used compared to the largest.

Measurement Matters - Time Travel

With all that history of measurement at his fingertips, how did it happen that Columbus was so far off in his estimates of the distance he had traveled west on his first voyage? The answer was probably not space, but time. To determine how far north or south you are, you can simply use the angles of certain known stars or constellations. Native American, Chinese, and Maori navigators had extensive knowledge of how to guide their travels by the stars. However, to determine longitude, an accurate clock is an extremely useful tool.

The divisions of time that we use today were proposed by Babylonians. Their units of time, bēru, were equivalent to 30^o of the movement of the sun each day. Clocks were made from sand or water.
 

Mini-STEM: Build a Clepsydra

Greeks, Egyptians, and many other societies built clepsydras (Greek for "water thief"), or water clocks, to keep time.

• Your challenge is to build a water clock that will tell time to the nearest 15 seconds and alert you at the one-minute mark.
• Your materials can include tape, disposable cups, a small bell, sticks, and other safe materials you can find.
• Calculate the accuracy and precision of your water clock compared to the timer on your phone.

While many clocks were invented over the centuries, they all had problems with both accuracy and precision. Hourglasses with sand, water clocks, candle clocks, sundials… none were completely reliable. Early spring-driven clocks ran fast when they were first wound, and slower as time went on. This might not have been a big problem if the question was, "When is dinner?" But the precision of your timing device made a big difference if it was used to calculate how far you’ve come on a voyage of thousands of miles.

Think back to Columbus. His method of navigation is called dead reckoning. He would drop a log behind the ship and measure the speed it floated away with a timer. But he had no accurate timer!

The first reasonably accurate chronometer that could be used on a ship was invented by John Harrison in 1781. He achieved a precision of about 0.1 seconds per day, accurate enough to predict the position of a sailing ship within a mile on a month’s voyage.

If he had had both Harrison’s clock and the calculations of Eratosthenes from 300 B.C.E., Columbus could have told his crew with great accuracy how far they had come and how far they were from "the Indies." But would he have wanted to? He might have ended up in a different place in the history books—at the bottom of a very deep and dense Atlantic Ocean.