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, based on the length of the day in 1900, 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.
At one time, the meter 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 meter has been replaced; now, the meter derives from the second, through the speed of light, which is, by definition,
8
2.99792458 * 10
meters 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 centimeters, 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 meter in terms of a light wavelength existed, but that definition was based on a line in the spectrum of cadmium: the length of the meter 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 meter 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 meter 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 meter, 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 meter 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 millimeters (as opposed to 2.2633485174 millimeters), 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 meter 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 millimeters 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 meter. 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 meter 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 centimeters.
Prior to 1964, the inch was defined in the U.S. on the basis that a meter was exactly 39.37 inches long, which led to the inch being about 2.540005 centimeters long, and in Britain the inch was 2.539997 centimeters in length. (One older reference gives the meter being 39.37079 inches, and the inch therefore being 2.539954 centimeters in length.)
I think it is unfortunate that they missed their chance to define the inch as being about 2.540002 centimeters in length, so that the diagonal of a square 152 inches on a side would be exactly 546 centimeters, or the diagonal of a square 273 centimeters 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 centimeters are somewhat unwieldy. However, as rulers measuring inches are often divided in tenths of an inch, one could relate 15.2 inches to 273 millimeters. 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 millimeter, defining the inch as about 2.5399946 centimeters would lead to a square 284 millimeters 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 centimeters, the ratio 152 inches to 273 centimeters approximates the square root of two with an error of 0.000078 percent.
The diagonal of a square 273 centimeters on a side is 386.0803025278548... centimeters, while 152 inches of 2.54 centimeters each are 386.08 centimeters, so there is an excess of 0.0003025278548... centimeters. 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... centimeters. 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 centimeters on a side is 5.0109929... inches in length. For comparison, the diagonal of a square 7 centimeters on a side is 9.8994949... centimeters 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 centimeters on a side, which is 55.7200143574999... centimeters, is very close to 10 centimeters (about the diagonal of 7 centimeters) plus 18 inches (about the diagonal of the other 32.4 centimeters) since 18 inches is 45.72 centimeters.
Since these are all even numbers, we can note that 9 inches plus 5 centimeters is approximately the diagonal of 19.7 centimeters.
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.
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 diameters 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 diameter 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 centimeter, 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
grams.
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.)
Also, here are the values of some relatively little-known English units. When the height of a horse is given in hands, each hand is 4 inches. In Britain, people's weights were usually quoted in stone rather than pounds; one stone is 14 pounds. Also, in the Troy system of weights, 24 grains make a pennyweight, with 20 pennyweights making up a troy ounce; in Apothecaries' weight, the grain and the troy ounce are still used, but now 20 grains make a scruple, 3 scruples (or 60 grains) make a dram, and there are 8 drams to the troy ounce.
Even more obscure is the division of the inch into 3 barleycorns, or 12 lines. With 12 lines to the inch, and 12 inches to the foot, lines were sometimes indicated with a triple prime, just as the third, one-sixtieth of a second, in angular measure was indicated.
Although they are available in most reference works, also little-known are the fathom (6 feet or 2 yards), the rod (5 1/2 yards, or 16 1/2 feet), the chain (66 feet, 22 yards, or 4 rods), and the furlong (660 feet, 220 yards, 10 chains or 40 rods), these latter units composing the mile, which is eight furlongs in length. A league is three miles long.
It might also be noted that while an acre is unusual for a unit of area in not being a square something-or-other, an acre is equal to ten square chains, and a square furlong is equal to ten acres. A unit used for measuring nuclear cross section, the barn, is 10^-28 square meters, or 10^-24 square centimeters, or 10^-22 square millimeters. Before looking it up, I had thought I had seen the figure 10^-15 relating it to a conventional area measure, which would have led nicely to the riddle "Why is a barn like an acre", but no such mistake was actually made in its definition. However, the millibarn and nanobarn are also used as units of cross-section, and these units are acre-like.
While the exact value of the historical cubit is variable, a cubit can be thought of as being roughly 18 inches in length, composed of 6 spans each 3 inches long, each span consisting of 4 digits each 3/4" wide. Three-quarters of an inch is also the horizontal spacing of keys on a typewriter.
Four cubits of 18 inches make a fathom of 6 feet; 100 fathoms make up the Greek stadium.
Another ancient foreign system of measure that is of some interest is that used in Japan.
When Japanese units were standardized in terms of metric units in 1891, the unit of length, the jo, was defined as 100/33 of a metre; the unit of volume, the sho, was defined as 2401/1331 of a litre, and the unit of mass, the momme, was defined as 15/4 of a gram.
There are ten shaku in a jo; ten sun in a shaku; ten bu in a sun, and ten rin in a bu. (The term bu is also a Japanese word for the fraction one-tenth, and rin also means one-hundredth.)
This is of importance today because the grid of a Go board is traditionally one shaku and four sun across, but one shaku and five sun long.
The spacing of the lines on the grid of a Go board is, of course, one-eighteenth the size of the whole board, and thus while it can be exactly expressed as a fraction, it cannot be so expressed as a decimal. This includes the thickness of the grid lines, of course, and thus the simplest calculation is based on the assumption that the outermost lines are the same thickness as the inner lines, and that the measurement of the size of the board is from the middle of the lines.
| Unit of measurement: | bu | millimeters | inches |
| Across | 140/18, or 7 7/9, or 7.777777... | 23 169/297, or 23.569023... | 0.9279143... |
| Along | 150/18, or 8 1/3, or 8.333333... | 25 25/99, or 25.252525... | 0.9941939... |
The lines on a Go board are about 1/25th of an inch, or one millimeter, in width. If we take this width to be, therefore, 0.3 rin, or 0.03 bu, and subtract it once in both directions, so that eighteen lines plus eighteen spaces are 1.3997 shaku across and 1.4997 shaku wide, we still do not get figures that divide exactly by eighteen in either direction.
Eighteen times 0.9941939... inches is about 17.89 inches.
For comparison, the American Checker Federation specifies that a checkerboard for tournament use is to have squares 2 inches on a side, making the board 16 inches long and wide. (The pieces are to be disks from 1 ¼" to 1 ½" in diameter.)
The USCF specifies that a chessboard as used in tournaments is to have squares from 2 inches on a side to 2 1/2 inches on a side, so that the length of the board may range from 16 inches to 20 inches. Note that 18 inches falls in the middle of this range.
Chess pieces, of course, vary in size, the King being the largest and the Pawn the smallest. Since the size of a chess set is usually specified by the height of the King, a standard for the size of chess pieces for tournaments is also given: the King height should range from 3 3/8" to 4 ½", and the base of the King should be from 40% to 50% of its height.
The same proportion of base to height is given by FIDE; one web site claimed that FIDE spacified a range of King heights from 8.5 cm to 10.5 cm, which is from just over 3 1/3" to just over 4 1/8", which leads to a FIDE size range that is centered about a smaller size of piece than the USCF range, although chess pieces with a King height from 3 ½" to 4" will be well within both ranges. The official FIDE web site simply says that the height of the King should be "about 9.5 cm", which is approximately 3 ¾". As for the size of the squares, their standard is from 5 cm to 6.5 cm, or just under 2 inches to just over 2 ½ inches, so this portion of the standard is essentially compatible.
The web site of one manufacturer of chess pieces notes that a chess board is well-matched to the size of the chess pieces used on it if the base of the King has a diameter which is 75% of the size of the square. Comparing this rule of thumb to the specifications noted above, the smallest possible King base is 67.5% of the size of the smallest square, and the largest possible King base is 90% of the size of the largest square.
This means both that the range of chess piece sizes is broader than that of board sizes, and that a board designed to be used both for chess tournaments and checker tournaments, with a set of chess pieces designed to go well with the board, is possible.
In the case of International Checkers, the Fédération Mondiale du Jeu de Dames does not define the size of the playing units in terms of exotic foreign units of measure such as shaku, or inches. Rather, the squares which make up the 10 by 10 board of this game are from 3.5 centimeters to 4.5 centimeters. The diameter of the checkers is to be from 7 to 14mm less than the size of a square, and their thickness is to be from 20% to 25% of their diameter. Also, they specify that a raised border of 5 mm to 6 mm is to surround the playing area of the board, which is quite an unusual requirement, unless the meaning is actually that the board itself is to have that thickness.
Thus, the board is from 35 centimeters to 45 centimeters in size; for comparison, 18 times 2.52525... centimeters is about 45.45 centimeters.
Since we have examined the game of Go, played on a board of 19 by 19 points and 18 by 18 squares, with the size of the board defined in the tradiditonal Japanese units of the bu and the shaku, and we have examined Chess, on the 8 by 8 board, and forms of Checkers on an 8 by 8 board and a 10 by 10 board, with both inches and centimeters used as units, and we have found that the size of the board always seems to be consistent, and in the neighborhood of 18 inches, it would seem to follow that wherever the adult human body is the same size, whatever system of units may be used, this is the size found to be most appropriate for a board used four tournament play.
The diameter of Mars at the Equator is now known to be about 6,786 kilometers; but if we took instead a value of 6,807.578 kilometers, this would give an equatorial circumference of 21,386.64 kilometers or 13,189.04 miles. Dividing this equatorial circumference into 360 parts would yield a karad of 36.914 miles; the haad, being the hundredth part of a karad, would then be 1,949.0592 feet in length, as it was given to be in the book A Fighting Man of Mars by Edgar Rice Burroughs. (In Thuvia, A Maid of Mars, an erroneous figure of 2,339 feet is given, and this has been often quoted. However, if the error was in assuming that on Mars the circle is divided into 360 parts, as is the case on Earth, then it might be noted that if the karad were one 300th part of the equatorial circumference of Barsoom, at 43.963 miles, then the haad would be 2321.271 feet in length.)
If one divides the haad into 200 parts, one obtains the ad, which would be 9.745296 feet; and dividing the ad into ten parts, one obtains the sofad, which would be 11.6943552 inches; and, finally, dividing the sofad into ten parts, one obtains the safad, which would then be 1.16943552 inches, or 2.9703662208 centimeters.
The FMJD dimensions for the squares of a 10 by 10 Checker board are from 3.5 centimeters to 4.5 centimeters; if we take 4/5ths of the dimensions provided by FIDE for the squares of an 8 by 8 Chess board, we get a range from 4 centimeters to 5.2 centimeters. Using the latter range as our primary standard, and attempting to convert it to a convenient multiple of a unit approximately 3 centimeters in length...
Until such time as the relevant Barsoomian authorities may be contacted directly, I shall humbly suggest that the regulation size for the squares of a tournament Jetan board should be about 1.5 safads (about 4.45555 cm, or 1 ¾ in.), with a permitted range from 1.35 safads to 1.7 safads.