The Incorporeal Ruler

By The Metric Maven

It has been noted that on Star Trek (TOS) that one never sees a ruler. Measurements are made, but only using a tricorder or some other device it appears. Rulers had somehow been banished to the past. Technological change has apparently rendered rulers obsolete in the 22nd Century.

One evening I was watching an episode of World’s Toughest Fixes. Sean Riley, the host and participator in the fixes was involved in dealing with a problem where the clearance of a roof, inside of a giant building, which was many, many meters high needed to be known to within 125 mm or so. This accuracy and precision over such a large distance was causing great heartburn. Riley reached into his complement of tools and produced one which could be placed on the floor and would measure the distance to the roof with millimeter accuracy. It uses a laser. I was dumbfounded. This was such a cool measuring device I wanted one, but could not really justify it in my line of work. After the measurement was performed, the crew was now confident they had enough clearance, even though none of them could have possibly used a tape measure to directly determine the unknown distance. The customary use of a graduated rule was essentially out of the question. This lead me to think about the origins of the common everyday ruler.

The earliest known graduated measuring rod dates from 2650 BC, and comes from ancient Sumer. It is very crude and has six graduated lines across its length. It’s not much to look at and is reproduced below.

Copper Alloy Sumer Cubit Bar (click to enlarge) -- Wikimedia Commons

One can see considerable scale refinement in a surviving cubit rod from Ancient Egypt. The rod has a number of equal divisions along its length. This base division has separate divisions of the base length division, into 1/2, 1/3, 1/4, 1/5 … to 1/16. This device is beginning to look remotely like a modern measurement scale, but still has a long way to evolve.

Cubit Rod from the Turin Museum (click to enlarge) -- Wikimedia Commons

Rectangular/square bars with fixed graduations continued in use with more precise markings and of more stable materials into the 19th Century.  A British Parliament fire in 1834 destroyed their length standards and new ones were commissioned to replace those lost. The British fabricated one inch square rods which are 38 inches in length. The rods are constructed using Baily’s metal No. 4 consisting of 16 parts copper, 2 1/2 parts tin, and 1 part zinc. A pair of markings on gold plugs, which are recessed into the bar, defined the length of one yard at 62° F. The plugs are recessed into the bar so they would be co-located with the axis of symmetry. This is known as the “neutral plane” where length error due to the sag of the bar under its own weight is minimum. Two protective plugs were used to preserve the defining lines. Despite the fact that the standard provided by the British was not sanctioned as an official version by parliament, only the first five copies were, the English Bronze Yard No. 11 was the official standard of the US until 1893, when the metric system was adopted.

The Official-Unofficial Yard Bar No. 11

The change to metric standards for length definition in the US occurred because the yard and its copies were shrinking at a rate of one part per million every twenty years. This shrinking was due to the relaxation of internal strain which had been introduced in the fabrication process. It was also noted that the pound provided by Britain was also “unfit for use.” By the time these problems were understood, it appears that the technical community in the UK saw the future was with metric and joined in the international collaboration. The best technical minds available were all focused on the best and most stable measurement artifact possible, which did not use a prohibitively large amount of platinum. The book The Evolution of Weights and Measures and the Metric System (1906) has an illustration of the candidate cross-sections which were studied for a meter length standard. The idea was to produce a very rigid bar which would create a very stable “neutral axis” in the center of the bar.

The use of an X cross-section was finally settled upon. Below is what was finally adopted.

There were two versions, one version with a pair of lines, and another which the length of a meter was defined by the ends of the bar. Below is a meter bar with engraved lines.

Meter Bar

This was the first adoption of a non-rectangular cross section measurement artifact for use as a calibration standard. The original meter bar was rectangular (figure 1). This change to a X design illustrates that the metric system continued to evolve, and the English standards atrophied and became neglected. T.C. Mendenhall (1841-1924), superintendent of the U.S. Coast and Geodetic Survey, realized that if the US wished to maintain accurate metrology when compared with the rest of the world, metric standards would have to be used. There was no alternative. Fortunately, the US had signed the Treaty of the Meter and a set of US metric standards existed and were available. Congressional inaction required Mendenhall to make a decision to use the existing metric standards, instead of the British ones, despite the fact that there was no legislation which sanctioned such a change. Mother nature does not succumb to the will of legislative bodies, and so Mendenhall now defined all lengths in the US in terms of the meter so that American metrology would not be left behind.

The neglected British measurement standards no longer kept up with improvements in metrology; they were now technologically dead. All new metrology improvements were only to be found within the metric system. Technical improvement continued to evolve until finally the meter artifact itself became obsolete and  was replaced with a length based on the counting of a number of wavelengths of light (pioneered by Americans). From this point forward, an artifact ruler was no longer the basis for a length standard, but was instead derived using a standard which was tied to a repeatable scientific phenomenon, which meant it could be reproduced anywhere on the planet–or even off the planet. Length definition had outgrown the primitive measuring stick used by our ancestors.

The laser measurement device used in World’s Toughest Fixes points to a possible future when rulers might become less ubiquitous (it is hard to imagine them actually disappearing from use). Unfortunately, we in the US continue to pretend there has been no improvement in measurement and its usage since the days when barleycorns were used to define length, and fractions were the state of the art for tape measures, rather than the use of decimals. It is an embarrassment for the US that our measurement usage has more in common with an ancient hand-carved cubit rod, than a precision laser.

Don’t Take Ye Olde English Wooden Nickels

Photo by the Author

By The Metric Maven

Bulldog Edition

I’ve always liked wood. My Grandfather was a carpenter, my father also knows how to work with wood. I, alas, do not. My father can generally recognize different woods, I haven’t a clue. The variety of woods that exist is astonishing. I recall reading about how the English carpenter and clockmaker John Harrison (1693-1776), used a wood called Lignum vitae, to construct wooden clocks. This wood actually has the property that is is self-lubricating, which allowed Harrison to make bearings and gears of it for his pendulum clocks. It is also one of the hardest woods in nature. In my youth I marveled at the Balsa wood rubber band driven airplanes available at the local dime store. I also have an interest in photographing wooden grave markers. An image of one of these “tombstones” I photographed is shown in the photo above.

I thought back on all this when paging through an old book called Science in Everyday Things by Engineer William C. Vergara and published in 1958. The book is a list of questions with answers, and one of them is: “Does all wood float?” What I assume he means is that given a solid cube of wood, will it float in water? When Vergara offers his answer, he informs his readers that the density of water is 62.5 pounds per cubic foot. Any wood with a density above this will sink and any below this will float. What struck me was the contrived and arbitrary nature of this value. It reminds me of the freezing point of water being defined as 32 degrees, an arbitrary magic number to be remembered on the Fahrenheit scale, because of poor measurement planning.

What struck me even more was that I was still going along with all the people who use kilograms per meter cubed. In my view, this is a vestigial Ye Olde English use of metric, which should be diminished. Density is the mass of an object divided by its volume. The everyday common volume unit allowed in metric is the liter, and a liter of water is essentially a kilogram. This means that a liter of water has a density which may be written as 1000 grams/liter. Both grams and liters are familiar units in the everyday world of an average person in a metric country. If a wood’s density is below 1000 grams/liter it will float, if it is above 1000 grams/liter it will sink. This is a nice Naughtin’s Laws friendly way to express water’s density, and it does not involve recalling a number like 62.5 pounds per cubic foot. This method essentially relates the specific gravities of the woods in an elegant fashion. It allows one to rationally list the densities of various woods in a way which one can immediately realize if they would float or not:

Wood             Density (g/L)

Balsa                           96
Yellow Pine                650
Maple                         704
Hickory                       816
Water Gum Tree      1000         (Density of water)
Black Ironwood        1040
Poison Ash              1104
Arapoca                   1200
Lignum Vitea           1229
Qeubracho              1393

Balsa is the lightest of the woods and a cube of it will clearly float in water. Yellow Pine, Maple and Hickory will also all float in water. Yellow pine was chosen for use in the caissons that were used to construct the Brooklyn Bridge.[1] Southern Yellow Pine was chosen for it’s ability to withstand large pressure and for the considerable amount of resin it contains. This makes it very resistant to rotting. Wood from the aptly named Water Gum Tree has neutral buoyancy, that is, it has essentially the same density as water and is compelled to neither float nor sink in it. A block of Water Gum Tree wood is like a helium balloon which floats at a stable position, neither rising nor falling to the floor. The word Quebracho means “Ax Breaker.” Given its high density, this name seems appropriate.

In his last sentence Vergara states: “Since wood weighing more than 62.5 pounds per cubic foot will sink, it can be seen that many  kinds of woods cannot possibly float.”

This is true for a single monolithic block of wood, but all these woods can be used to make vessels which will float, they only need to displace enough water to do so. The Civil Engineering students at Iowa State University each year create a concrete canoe. The density of concrete? It’s 2400 grams per liter. But concrete is not nearly as dense as steel which is about 8000 grams per liter. The hollow interior of a canoe or ship decreases the overall density of the ship enough to bring it well below the 1000 grams/liter threshold, which in turn allows it to float. Allowing for shaping, all wood will float.

I very much encourage the use of grams/liter for expressing density. I completely discourage the use of the cgs leftover, grams/cubic centimeter. In the case of steel its density is 8.05 g/cc. I also mostly tend to discourage the use of kilograms per cubic meter, as the units are out of the range of everyday measurement experience. It is, however,  very easy to convert from kg/m3 to g/L. For instance, the density of steel is 8000 kg/m3 which is 8000 g/L. The conversion factor is one. The numerical values are the same, just change the units. It’s just that easy, as the metric system is, and should be, when it’s employed in an articulate manner. People who insist on using Ye Olde English units like pounds per cubic foot—are just dense.

[1] The Great Bridge, David McCullough, Simon and Schuster, New York pg. 174-175.