On this page, we will take a brief look at some of the complications encountered when attempting to define units of measurement with high accuracy.
The second is defined as the time taken by 9,192,631,770 oscillations of the microwave radio frequency produced by an atom of Cesium-133 when the electrons in that atom are in the ground state, except for one that has emitted this radiation by making the transition from the upper hyperfine level of this state to the lower one. This definition was chosen to make the length of the SI second the same as that of a second of Ephemeris Time and so when this second began to be used in civil timekeeping (the changeover to "atomic time"), the use of "leap-seconds" became necessary immediately.
The 1952 definition of the second of Ephemeris time was based on the instantaneous length of the tropical year being 31,556,925.9747 seconds at the beginning of the year 1900, which was considered to be Noon GMT of December 31st, 1899, but according to Ephemeris Time rather than civil time. However, while the length of the year at that time was the basis for the standard, the length of the second was not 1/86,400 of the length of the day at that time, but was rather based on the average of the length of the day from 1750 to 1892 as previously worked out by Simon Newcomb.
At one time, the metre was defined as 1,650,763.73 wavelengths of the orange red line in the spectrum of Krypton-86 which corresponded to the radiation emitted by an electron moving from the orbital
2 p to the orbital 5 d 10 5
in an unperturbed fashion.
However, a scientist proceeded to measure the speed of light by performing an accurate measurement of the ratio between the wavelengths (and/or frequencies) of these two types of radiation. In Zen-like fashion, this convinced those responsible for the standards of the fundamental absurdity of the situation, and so now the definition of the second stands, but the definition of the metre has been replaced; now, the metre derives from the second, through the speed of light, which is, by definition,
8 2.99792458 * 10
metres per second.
This leads to the length of a metre being approximately 30.6633189885 wavelengths of the Cesium-133 microwave radiation noted above: the fact that the wavelength is not a tiny fraction of a metre is why it had previously been considered more suitable to a standard of time, for which a radio frequency which can be manipulated by electronics is more accessible than an optical frequency, than one of distance.
The speed of light in a vacuum can also be given in units of the uniform US/British inch of 2.54 centimetres, which leads to light travelling 186,282 miles, 698 yards, 2 feet, and 5 21/127 inches every second in a vacuum.
Note that it is more convenient to measure the length of waves of light through interference fringes, and the time between oscillations of radio waves through electrical circuitry, however. So the dual definitions allowed both time and distance to be more accurately defined.
Incidentally, as far back as 1927, a definition of the metre in terms of a light wavelength existed, but that definition was based on a line in the spectrum of cadmium: the length of the metre was defined as 1,553,164.13 times the wavelength of the 6436.4696 Ångström cadmium red line.
A. A. Michaelson had found the cadmium red line to be particularly monochromatic; he had made a measurement of the International Prototype Metre in 1892 which indicated the length of the metre as 1,553,163.5 times the wavelength of the cadmium red line, the wavelength of which was taken to be 6436.472 Ångström units.
The figure used in the 1927 standard of 1,553,164.13 wavelengths per metre corresponds closely to a wavelength of 6436.4696 Ångströms for the cadmium red line; that the cadmium red line is exactly 6436.4696 Ångström units long was adopted as a standard for the definition of the international Ångström unit in 1907, based on measurements in 1906 by Benoit, Fabry, and Perot of the length of the metre in terms of the cadmium red line.
Cadmium, however, has eight stable isotopes, the most common of which, Cadmium-114, has a natural abundance of 28.73%. so the accuracy of a standard based on natural cadmium would be limited by the small variations in the wavelengths of the same spectral line between different isotopes.
Prior to the use of Krypton-86 for the standard length of the metre, another possibility that was considered was to use Mercury-198. This isotope of mercury was created in 1946 by bombarding gold with neutrons, as gold, like aluminum, has only one stable isotope, and this method of creating isotopically-pure mercury was simpler than attempting to separate the isotopes of mercury by their miniscule difference in weight. Its 5460.753 Ångström spectral line was the one considered for use as a standard. Also, a current secondary standard for the metre is an iodine-stabilized helium-neon laser, the light from which has a wavelength of 6329.9139822 Ångströms.
It is necessary at this point to add that 3515.3502 wavelengths of the cadmium red line would, by the 1927 definition, subtend some 2.2633475317 millimetres (as opposed to 2.2633485174 millimetres), which seems to indicate a discrepancy in the definition of the Potrzebie. As a little arithmetic shows that this definition of the Potrzebie would lead to a metre of 1,553,163.45 wavelengths of the red line of cadmium in length, it seems apparent that Donald Knuth, then a 19-year old high school student, had converted from millimetres to wavelengths of cadmium light using a reference giving the 1892 figure.
In my efforts to sort out the mystery of that discrepancy, which led me to finding out about the 1892 figure, I had encountered a biographical essay on A. Michaelson by Robert A. Millikan giving 6438.472 Ångströms as the wavelength of the red line of Cadmium as determined by Michaelson which would indeed lead to a standard of about 1,553,163.5 wavelengths per metre. In the essay, the resulting standard as 1,555,165.5, which I took to be a misprint. Thus, the mystery of the difference between the length of the Potrzebie as defined in terms of the metre and that as defined in terms of the cadmium red line appeared to be solved, and further searching led me both to confirmation that 1,553,163.5 was the figure arrived at by Michaelson in 1892 and to find that the later standard derived from the 1906 measurements of Benoit, Fabry, and Perot as noted above.
It may also be noted that in 1964, an agreement was reached between the U.S. and Britain to define the inch as 2.54 centimetres.
Prior to 1964, the inch was defined in the U.S. on the basis that a metre was exactly 39.37 inches long, which led to the inch being about 2.540005 centimetres long, and in Britain the inch was 2.539997 centimetres in length. (One older reference gives the metre being 39.37079 inches, and the inch therefore being 2.539954 centimetres in length.)
Or is the British inch 2.539996 centimetres? There are sources giving both values for the British inch. I found the 2.539997 figure in the Encyclopedia Britannica article on weights and measures, while the 2.539996 figure occurs in news stories from 1938 and 1958 about the ongoing negotiations to establish agreement between Britain and the United States on the inch of 2.54 centimetres, so I would have been inclined to believe the 2.539997 figure to be the more accurate one. Yet another source gives the length of the inch based on the British "Bronze Yard #11" to be 2.53999944 centimetres.
Unlike the case of the U.S. inch, where the inch was defined so that 39.37 inches equalled one metre, so that the U.S. inch must therefore be 2.54000508001016002032004064.. centimetres, the British inch was defined by a separate physical standard, so there is no inherent exact ratio between the British inch and the metre.
A 1928 scientific paper, titled "A New Determination of the Imperial Standard Yard to the International Prototype Metre", by Sears, Johnson, and Jolly, gave the ratio of 1 metre to 39.370147 British inches, which gives rise to an inch of 2.5399955596... centimetres. Incidentally, the International Prototype Metre was made in 1872, while the Imperial Standard Yard dates back to 1855.
An earlier measurement, from 1895, gave the length of the metre as 39.370113 British inches, leading to a British inch of 2.53999778969... centimetres. That rounds to 2.539998, and so it cannot alone be the source of the other value; despite the fact that one source notes that the 39.370113 figure was the de facto standard for the length of the British inch for many years.
The Imperial Standard Yard is made from a bronze alloy, 82% copper, 13% tin, and 5% zinc, and it is defined to have its standard length at a temperature of 62 degrees Fahrenheit; the International Prototype Metre is made of 90% platinum and 10% iridium, and its length is valid at 0 degrees Celsius (32 degrees Fahrenheit, or the freezing point of water).
Incidentally, the Imperial Standard Yard was noted as having shrunk by one part per million in a span of 20 years, calling its accuracy into question.
It may be noted that the Pyramid Inch was claimed to be 1.00106 English inches, so that would make it about 2.5426894 centimetres long. The Pyramid inch was said to be 1/25th of a royal cubit.
In fact, an ordinary cubit, about 18 inches long (so they were at least right that cubits related better to Imperial measure than to the metric system) was divided into six spans (each three inches long), which were in turn divided into four digits (each 3/4 of an inch; and, indeed, the keys on our typewriter keyboards have 3/4 of an inch spacing even today). A royal cubit is seven spans instead of six, and so, nominally, it should be 21 inches long, but then standards of measure were less accurate in those days. The royal cubit was used in the construction of the Great Pyramid; thus, its sides had a rise of one royal cubit for a run of five and one-half spans; which, multiplied by four, gives 3 1/7, giving the appearance that pi is involved in the construction of the Pyramids.
In fact, though, serious archaeologists and historians now know that the Egyptian royal cubit was about 52.6 centimetres (give or take 3mm), or 20.7 inches - so it was indeed near to 21 inches, and actually somewhat smaller, and thus not 25.0265 inches long. To the extent, therefore, that such a thing as an ancient Egyptian inch has any meaning, therefore, it would be about 98 4/7 percent of an inch, not 1.00106 inches, in length.
While the Egyptian measure relates to Imperial measure by a factor of about .986, the currently accepted value for the Roman foot makes it .971 feet long; thus, while the foot grew on its way to Britain, in the middle, as it passed through Rome, it shrank. And the Romans did divide the foot into 16 digits as well as 12 inches, and so linking the cubit to the inch as I have done is legitimate.
I think it is unfortunate that they missed their chance to define the inch as being about 2.540002 centimetres in length, so that the diagonal of a square 152 inches on a side would be exactly 546 centimetres, or the diagonal of a square 273 centimetres on a side would be exactly 152 inches. After all, supporters of the metric system have always criticized the Imperial system as irrational; and it would be convenient if having two systems of measurement allowed one, by using both of them, to measure exactly both the sides of a square and its diagonal.
Of course, lengths of 152 inches and 273 centimetres are somewhat unwieldy. However, as rulers measuring inches are often divided in tenths of an inch, one could relate 15.2 inches to 273 millimetres. But inch rulers are more often divided into sixteenths of an inch.
If one were to use the same method to relate the sixteenth of an inch to a millimetre, defining the inch as about 2.5399946 centimetres would lead to a square 284 millimetres on a side having a diagonal of 15 and 13/16 of an inch.
However, failing changes in our systems of measurement, one can always simply make use of the fact that 20 squared is 400, 21 squared is 441, and 29 squared is 841 to come reasonably close to a 45 degree angle and still use exact distances.
Also, even if redefining the inch is excluded, if only a single unit of measurement is used, comparable ratios to approximate the square root of two would be 239:169 and 577:408, which are in error by 0.000875 percent and 0.000150 percent respectively, while, using the inch of 2.54 centimetres, the ratio 152 inches to 273 centimetres approximates the square root of two with an error of 0.000078 percent.
The diagonal of a square 273 centimetres on a side is 386.0803025278548... centimetres, while 152 inches of 2.54 centimetres each are 386.08 centimetres, so there is an excess of 0.0003025278548... centimetres. The older U.S. inch, such that 39.37 inches equal one metre, is still in use for survey purposes, and this inch is equal to 2.54000508001016... centimetres. If, of the 152 inches of the diagonal, 59 and 9/16 of those inches were measured using the older U.S. inch, and the other 92 and 7/16 of those inches were measured using the current inch of 2.54 cm, an even closer approximation to the square root of two would be obtained.
A more approximate measurement of the diagonal of the square can be obtained using much simpler numbers. The diagonal of a square 9 centimetres on a side is 5.0109929... inches in length. For comparison, the diagonal of a square 7 centimetres on a side is 9.8994949... centimetres in length. The discrepancy, in addition to being in the opposite direction, is very nearly 3.6 times as large. So, the diagonal of a square 39.4 centimetres on a side, which is 55.7200143574999... centimetres, is very close to 10 centimetres (about the diagonal of 7 centimetres) plus 18 inches (about the diagonal of the other 32.4 centimetres) since 18 inches is 45.72 centimetres.
Since these are all even numbers, we can note that 9 inches plus 5 centimetres is approximately the diagonal of 19.7 centimetres.
Since those words were written, it occured to me that there might be another irrational number that should instead be used to define the relationship between the centimetre and a modified inch.
If one decides that a computer keyboard should be tilted at an angle of ten degrees, then while the spacing of circuit traces for the different keys would be, horizontally, three-quarters of an inch, vertically they would be that distance divided by the cosine of ten degrees. So, if that could be made something reasonably easy to specify...
With an inch of 2.54 centimetres, the distance in question is 1.93438769564234419687... centimetres approximately. If, for example, we want that distance to instead be 1.9344 cm exactly, we would have to change the inch to 2.540016156569... centimetres.
On one page on this site, I note that the Egyptians had a royal cubit of seven spans as opposed to the regular cubit of six spans. But I note at least one page that claims the original form of the royal cubit in Egypt was a measuring unit for measuring diagonals.
The U.S. pound was redefined as 453.59237 grams in 1959, having previously been defined on the basis of 2.2062234 pounds equalling one kilogram exactly, leading to a pound of about 453.5924277 grams, nearly identical to the British pound.
A pound is 7000 grains in weight, that is, the normal, 453.59 gram pound used to weigh food. Thus, if you have peas to weigh, you use this pound, which is called the avoirdupois pound. The troy ounce, which is used to weigh gold, however, is 480 grains in weight, and there are twelve troy ounces in a troy pound.
Thus, the relevant conversions are:
Grains Grams Avoirdupois Pound 7000 453.59237 Troy Pound 5760 373.2417216 Troy Ounce 480 31.1034768 Avoirdupois Ounce 437.5 28.349523125 Grain 1 0.06479891
And so we can make this chart of ounces versus grams for some common chocolate bar sizes:
Ounces Grams 1 28.35 1 1/2 42.52 1.58733 45 1 3/5 49.36 1 3/4 49.61 2 56.7 2 1/2 70.87 2.64555 75 3 85.05 3 1/2 99.22 3.5274 100 4 113.4
Here is a rough table of some values for gold, and the relative value of a dollar in terms of gold under those values:
-1834 -1933 -1971 2011 Up to 1834: $19.38 $1.00 $1.07 $1.81 $81.80 Up to 1933: $20.67 94¢ $1.00 $1.69 $76.69 Up to 1971: $35.00 55¢ 59¢ $1.00 $45.29 January 21, 1980 $850.00 2.28¢ 2,43¢ 4.12¢ $1.86 April 2, 2001 $255.95 7.57¢ 8.08¢ 13.67¢ $6.19 July 13, 2011: $1,585.20 1.22¢ 1.3¢ 2.2¢ $1.00
The peak high in 1980 (New York), the peak low in 2001 (London), and the new record-setting price (New York) are noted.
People writing mediaeval role-playing games should note that due to the difference in density between gold and silver, if one uses the historic ratio that gold is 16 times more valuable than silver by weight, then it is also (approximately) 25 times more valuable than silver by volume. This will make it possible to determine more accurately how much treasure your characters can carry in their backpacks, while still using handy round numbers.
Prior to the devaluation of the U.S. dollar in 1933, which also led to the abolishment of gold coinage, the U.S. dollar was defined by defining the Gold Eagle, a coin with a denomination of $10, as being composed of 258 grains of 9/10 fine gold. In U.S. coins, the remaining tenth was made up of half silver and half copper, which were not without value themselves.
The Constitution originally defined the dollar as 371.25 grains of silver, and that is 90% of 412.5. In 1872, the weight of the silver dollar was reduced to 384 grains of 9/10 fine silver, because the price of silver had gone up again relative to that of gold, and gold was generally accepted as the international standard of monetary value. However, unlike the previous silver dollar, while this coin was still intended to be worth nearly a dollar, it was not expected to be worth exactly a full dollar; it was not token coinage, but it was subsidiary coinage.
If the situation between 1834 and 1872 is used as the basis, then, an attempt can be made to determine the exact nominal value of pure gold in U.S. dollars.
A gold dollar of 25.8 grains of 9/10 fine gold consists of 23.22 grains of pure gold, 1.29 grains of pure silver, and 1.29 grains of copper.
371.25 grains of pure silver and 41.25 grains of copper equal a dollar.
Thus, 1.29 grains of pure silver and 0.14333... grains of copper equal 0.3474747... cents.
So, 23.22 grains of pure gold and the remaining 1.15333... grains of copper equal 99.6525252... cents; thus, if we neglect the value of the copper, that works out to one troy ounce of gold (480 grains) being worth almost exactly $20.60 instead of $20.67.
If silver were worth 1/16th as much as gold, this would make 371.25 grains of silver worth about 99.58 cents instead of a dollar.
In 1805, the price of copper was considered high at 138 pounds a ton, or about $690 a ton. A ton being 2000 avoirdupois pounds of 7000 grains, this would be about 2.366 cents per troy ounce, compared to about $1.29 per troy ounce for silver and $20.60 per troy ounce for gold. Thus, the ratio of value by weight, historically, had been in the neighborhood of 50 to 1 between silver and copper, or perhaps higher.
Given the composition of the Gold Eagle above, a half-eagle, with a $5 denomination, would be composed of 116.1 grains of pure gold, 6.45 grains of pure silver, and 6.45 grains of pure copper. A Gold Sovereign from Britain, on the other hand, is composed of 113.0015 grains of pure gold, and 10.2729 grains of copper, for a total weight of 123.2744 grains, or 7.9881 grams. Another source gives the weight of a British Gold Sovereign as 5 pennyweights, 3 171/623 grains; one pennyweight is 24 grains, so this is 123 171/623 grains, or approximately 123.2744783306581 grains. This works out to about 113.0016 grains of pure gold at 22 carats.
If the halfpenny were exactly equal to the cent, making the British pound equal to $4.80 in value, 113 grains of gold to the pound would imply about 117.7 grains of gold to five dollars; thus, while a pound was not worth quite as much as five U.S. dollars, it was actually worth about $4.85 at the time of the gold standard. (Actually, the value of the mint par of exchange used at the time was $4.8665.) While the Canadian dollar was made to be equal to the U.S. dollar, the Newfoundland dollar was about 1 1/3 cents more than a U.S. dollar, as one would expect from a dollar equal to 100 ha'pennies.
In that era, other countries also had currencies which were fixed in value in terms of gold, and thus to one another. Thus, 19.2952 cents, at 4.48036 grains of gold, was the value of a French Franc, a Swiss Franc, a Belgian Franc, an Italian Lira, a Bulgarian Lev, or a Greek Drachma in those days.
Here is a rough table of coin diametres and values, where the values are in terms of the value of U. S. currency in precious metals between 1892 and 1933, and the upper value is for silver coins, the lower for gold coins:
13mm 15mm 16mm 17mm 4 5 6 1/4 7 13/16 $1.00 $1.25 $1.56 1/4 $2.45 5/16 17mm 18mm 19mm 20mm 8 10 12 1/2 15 5/8 $2.00 $2.50 $3.12 1/2 $3.90 5/8 20mm 22mm 24mm 26mm 16 20 25 31 1/4 $4.00 $5.00 $6.25 $7.81 1/4 26mm 28mm 31mm 33mm 32 40 50 62 1/2 $8.00 $10.00 $12.50 $15.62 1/2 33mm 35mm 38mm 41mm 64 80 $1.00 $1.25 $16.00 $20.00 $25.00 $31.25
Moving one column to the right increases the weight and value of the coin by a factor of 1.25, moving one row down doubles the weight and value of the coin. The sizes have been formed on the basis of the currencies of several different countries, and thus they tend to be smaller than the sizes of U.S. coins, and/or larger than the sizes of British coins of corresponding value (1 shilling = 25 cents, 1 florin = 50 cents, 1 crown = $1.25, 1 pound = $5.00) up to the year 1919.
Because copper or bronze coins tend to be token coinage, their size is quite variable; the value ratio of 20:1 for gold and silver at the same diametre is consistent, but that between silver and copper can vary from 25:1 to 50:1.
The density of gold is 19.32 grams per cubic centimetre, that of silver 10.5 grams/cc, that of copper 8.96 grams/cc. A ratio of 64:1 in value by weight would lead to a ratio of 75:1 in value by volume for silver and copper as metals. Copper coins tend, however, to be a token currency, that is, a metal form of paper money as it were, and so the ratio in weight of copper coin to silver coin would be more like 16:1 or 32:1.
The atomic weight of gold is 196.9665, and its one stable isotope is Gold-197. An atomic mass unit is
-24 1.6605655 * 10
Thus, one grain of gold is .064798918... grams of gold, and would contain some
20 1.9811592 * 10
atoms of gold.
Using a 16:1 ratio of value between gold and silver, and a 64:1 ratio of value between silver and copper for a 1024:1 ratio of value between gold and copper, one finds that U.S. gold coins have a metal content whose value is that of
25.8 * (.9 + .05/16 + .05/1024) = 24.038847656...
grains of gold per dollar, one could, if one wished, define a dollar as the pecuniary value of
21 4.7624784 * 10
atoms of Gold-197, or, as they say,
197 Au 79
contained in a good delivery gold bar, which is a bar of gold that is at least 99.5% fine, and which has a mass of approximately 400 troy ounces, ranging from 350 troy ounces to 430 troy ounces (the actual fineness and the weight being marked on the bar). 400 troy ounces is about 12,441.4 grams, and the range would be from a low of 10,886.22 grams to a high of 13,374.49 grams.
(Whether or not the avoirdupois pound is a unit of force instead of a unit of mass, the troy ounce is definitely a unit of mass.)
Speaking of the kilogram, I have read news items noting that the international standard kilogram has experienced changes in its mass, and that an effort was underway to make a more accurate standard from a sphere of silicon.
The metre and the second can be defined in terms of spectral lines or microwave emissions from specific substances. Could the gram be physically defined?
Originally, the litre was defined as the volume of a kilogram of water. As well, a litre was nominally 1,000 cubic centimetres of water. However, because water has a pronounced tendency to dissolve, at least to a limited extent, almost any other substance, it is not a good material to use for a physical standard. Thus, for a time, a litre, instead of being exactly 1,000 cubic centimetres or one cubic decimetre, was instead officially defined as 1.000028 cubic decimetres, because this was, as near as could be determined, the volume that a kilogram of water actually occupied; this anomaly was corrected in 1964, and thus now one millilitre (ml) is exactly the same as a cubic centimetre (often noted by the non-SI abbreviation cc instead of cm3).
Thus, if it were desired to give the kilogram a reproducible physical definition based on water, it would have to be redefined so that the new kilogram was 0.999972 existing kilograms, and this would not be acceptable, being too large a change, and affecting the derived units of energy and force, and hence the various electrical units.
Liquid measure has always been one of the more confusing areas of measurement.
In the United States, a gallon is 231 cubic inches by definition.
In the British imperial system, a gallon was originally defined as the volume of ten pounds of water. Subsequently, the British gallon was defined in terms of metric units as 4.54609 litres; this was in 1985, after the litre had become exactly a cubic decilitre.
The Imperial gallon is larger than the U.S. gallon, but the Imperial fluid ounce is smaller than the than the U.S. fluid ounce, due to one difference between the two systems:
U.S. Customary British Imperial 1 gallon = 4 quarts = 3.785411784 litre 1 gallon = 4 quarts = 4.54609 litre 1 quart = 2 pints = 946.352946 ml 1 quart = 2 pints = 1.1365225 litre 1 pint = 2 cups = 473.176473 ml 1 pint = 2 1/2 cups = 568.26125 ml 1 cup = 8 fluid ounces = 236.5882365 ml 1 cup = 8 fluid ounces = 227.3045 ml 1 fluid ounce = 8 drams = 29.5735295625 ml 1 fluid ounce = 8 drams = 28.4130625 ml 1 tablespoon = 4 drams = 14.78676478125 ml 1 tablespoon = 4 drams = 14.20653125 ml 1 teaspoon = 4 scruples = 4.92892159375 ml 1 teaspoon = 4 scruples = 4.73551041667 ml 1 dram = 3 scruples = 3.6966911953125 ml 1 dram = 3 scruples = 3.5516328125 ml 1 scruple = 1.2322303984375 ml 1 scruple = 1.18387760041667 ml
the space in the table dividing the part in which the U.S. units are smaller from the part in which the U.S. units are larger. Often, the tablespoon is approximated by 15 ml in recipies now, although I've read that in Australia, 20 ml is instead used as the approximation. Thus, one could speak of a metric scruple of 1.25 ml.