In many television programs about mathematics that involve weights and measures, one is often taken to an open air market. The presenter will immediately seize upon the utility of numbers which have numerous divisors. The number twelve will be immediately enlisted. If one has a dozen eggs, then it can be divided up by 1, 2, 3, 4, 6 and 12. Often they move on to describe the amazing number of ways that 60 may be divided: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, which is why one has clocks with 60 seconds in a minute, and 60 minutes in an hour. One can imagine oranges, apples, pears and such all being sold in integer groups. Often it has been my experience that a person can purchase any of these fruits in any number they wish.
When one considers purchasing walnuts, they are small enough that counting them out begins to tax one’s time. It is still possible, but selling them in 60 walnut quantities takes time to count out. It also takes time for the purchaser to count them out, and make certain that all 60 walnuts are in a given bag.
Wheat is a commodity that like oranges, eggs and walnuts, exists in integer units, but the individual grains are so small that the amount of time needed to count out 7000 of them, which was the definition of a pound, is prohibitive. Do my seven thousand wheat grains each have the same mass as those used to define a “grain“? Counting out seven-thousand grains definitely takes a lot of time, and checking each one against a “standard” grain would be untenable. Of course, one could count out 7000 wheat grains and then use a balance to compare a bag with 7000 grains to one which you are pouring into a second sack. When the balance is level, a naive consumer might assume that the two bags contain exactly the same number of grains. Who is going to take the time to count?
On closer examination, one knows that the reference bag has 7000 grains, but because of the variation in the masses of individual grains, perhaps because they came from a location far away in a country with different growing conditions, the new bag might contain more than 7000 grains, they are just smaller, and each possess less mass. This is the beginning of the idea of measurement, versus the notion of counting. People seem to realize that the same amount of “stuff” is in each bag if they balance, even if the individual grain count does not match. The question is, who’s bag of 7000 grains should be the one used by everyone as a standard? This is where the modern notion of measurement begins to appear.
One can’t be certain that the number of grains in all the bags are equal to the seven-thousand in the “standard” bag, but instinctively people seem satisfied that the same “amount” of wheat has been meted out.
Robert Hooke (1635-1703) was the first to note that the length of a spring, within limits, is directly proportional to the force of an object which hangs from it. We can take our 7000 wheat grains, hang them from a spring which obeys “Hooke’s Law” and use the length the spring stretches, using our standard, as a known “calibrated” point. In the case of a spring we could put a pointer on the spring, and then place a mark at zero, when no grains are being measured, and a mark at 7000 wheat grains. A graduated scale can be placed behind the pointer. The location of the pointer is no longer restricted to single units of grain, it can point to an infinite number of locations along the scale distance from zero to seven thousand wheat grains. The divisions on the scale can be subdivided at will to produce more and more precision. We have stopped counting, and have begun to measure.
We can define seven-thousand wheat grains in terms of an indirect abstract quantity, not attached to a specific concrete item, such as cloth, grain or wood. This proxy quantity of “general stuff” we call an avoirdupois pound. The pound can in turn be used as a reference amount for a measurement of the quantity of any substance, corn, wheat, fish, bird seed or whatever. A person can fabricate a metal object which deflects the measurement pointer by the same amount as the wheat grains which make up a pound so that we can have a more stable, reproducible, and reliable standard. A second check can be accomplished by using a balance to make certain the two objects, the grains of wheat and the piece of metal, have the same amount of “stuff” in them. We call this abstract amount of stuff “mass” these days. So now we have created a one-pound mass for a standard, and we can measure commodities to as much of an exactness as we can produce graduations for the pointer to point at, and resolution for our eyes to read.
Once again, it is a problem to decide whose bag of wheat grains is used to determine which piece of metal is considered a pound. The history of weights and measures is generally a history of fraud and deceit. The definition of a standard value of mass, was not very standard, and variations could be used to cheat when trading. Below is a table of all the competing standards for a pound that I could locate:
So what do we do? Well, John Wilkins (1614-1672) originally defined his unit of mass, which would later be known as the Kilogram, as a cube of water with sides which are one-tenth of of his base unit. This base unit, with a different definition, would later be known as the meter. In other words, a cube of water with 100 mm sides is the original mass standard for the metric system. A cube of pure water, at a given temperature, made sense, but again, temperature could affect this definition. The temperature of water’s maximum density was chosen as a calibration point. When the value of this mass was determined by the French, during the development of the metric system, it was preserved in a more practical way, as an equivalent mass of platinum-iridium alloy. The relative of this agreed-upon mass is the International Prototype Kilogram (IPK).
The point of measurement, versus counting, is that it produces a continuum of available measurement values, and this value is independent of integer, or discreet values of poppy seeds, wheat seeds, barleycorns, bird seed or anything else. Once one has an agreed upon unit of mass, such as the Kilogram, it may be indefinitely subdivided. An easy way for humans to subdivide this base value, is by using 1000’s. The measured value is found on a continuum of available values, which can be further divided if needed. This is not counting by any stretch of the imagination. It is measurement. The argument for a choice of a numerical base which has lots of divisors is of no import when you have a continuum of possible measurement values.
So is the idea of using numbers which have lots of divisors irrelevant to the metric system? No, they are only irrelevant to metric system measurement. When metric units are chosen such that the amount of precision needed for everyday work is slightly smaller than required, integer values again become important. What I mean by this can be illustrated with metric housing construction in Australia and the UK. In order to make the description of lengths easy, we choose a unit length which in all practical circumstances will always be an integer. The unit chosen for construction is the millimeter. The millimeter is small enough that one never needs to use a decimal point in everyday construction. We have chosen to go back to integers (simple whole “counting” numbers). This is converting measures back to countable “atoms” of measure.
We use our modern measurement system to define a small length value, the millimeter, which is solidly known, rather than using a pre-metric small unit which varies—like a wheat or barlycorn grain. When we use this small unit to produce integers, we can use convenient values which indeed have lots of factors for division. In the case of metric construction, the value chosen is 600 millimeters for stud spacing. Its factors are: 1 2 3 4 5 6 8 10 12 15 20 24 25 30 40 50 60 75 100 120 150 200 300 and 600. What we are doing is not exactly measurement when we construct a house, it is equating multiples of integer values with multiples of a measured integer value, which is a different exercise. When we do this, it makes perfect sense to choose lots of divisors. With millimeters we have “atomized” the values on the construction drawings we are using to guide us. If we want to add in features, such as a window, not originally present on the drawing, or when initially creating a drawing, chances are that we will be able to divide the newly inserted distance easily. This is because of the conscious choice to use small units which can remain integers. We are not measuring in this case, we are back to counting.
Of course as we spent more time measuring our world, we discovered that it is actually discontinuous when it comes to fundamental values of mass. John Dalton (1766-1824) realized and demonstrated that the world is made of atoms. Each individual atom has a defined mass, but the same type of atom can have a range of masses. For instance, tin has atoms that are all chemically tin, but possess ten different mass values. These different mass variations of chemically identical atoms are called isotopes. Tin has ten isotopes, cesium has thirty nine!
One of the candidates to replace the current Kilogram standard, which is still an artifact from the nineteenth century, is the silicon sphere. This is a sphere of silicon atoms that will contain a known number of them. If a person knows the mass of each atom in the sphere, and their total number, it can be used to define a mass. In strange way, this procedure is similar to using 7000 wheat grains, but in this case we know that if an atom of silicon is of the same isotope as all the others in the sphere, it possesses a mass which is identical to all the other silicon atoms present. One of the largest difficulties for the team which is attempting to make a silicon sphere Kilogram mass standard, is making certain that all the silicon atoms present within the sphere are of the same isotope. Silicon 28 is the chosen isotope the silicon sphere team will use to create a new Kilogram standard—after counting all the atoms of course. We are counting an integer number of atoms, so that we can develop more accurate continuous set of measurement values, just as was done in the past with wheat grains. These values, which are continuous subdivisions of mass when compared with the discreet values of the atoms in the standard, may be used for the measurement of values which are smaller than the silicon atoms used. But remember, counting is not measuring.