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Standard Wheat

Standard Water

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Various Metrological Water Units

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grain/in³ 888888888888888	g/dm³ 888888888888888	Description
250.000 000 000 000	988.568 025 364	250 grain/in3
251.999 879 452 198	996.476 092 888	66 prussian lbs (@ 466.771 gms) / ft3 (139.13/443.296) UI
252.000 000 000 000	996.476 569 567	252 grain/in3: base 28 thingie
252.057 613 168 724	996.704 387 713	1680 lb = 1 cu yard
252.081 345 570 296	996.798 232 086	60 french lb = 12.16-inch cube
252.101 429 857 562	996.877 650 823	198 gt/cyl in
252.108 991 618 422	996.907 552 083	TFPS water = 6125/6144 metric water
252.273 899 939 860	997.559 644 458	300 lb / 20.2666666-inch cube
252.276 386 920 435	997.569 478 656	TILF water = 0.997984² metric water
252.290 609 521 828	997.625 650 950	Mohr cc, used in chemistry
252.324 994 593 036	997.763 726 631	100 lb = 10 UK gallon [G2 ~ 277.420 in³]
252.457 934 874 658	998.287 368 666	100 lb = 10 UK gallon [G1 ~ 277.274 in³]
252.499 487 274 449	998.451 678 149	2240 lb = 39.6 inch cube
252.525 252 525 525	998.553 056 974	100 lb = 12 US gallons
252.583 528 491 081	998.784 000 000	ILF water = 0.997984 metric water
252.626 893 796	998.955 478 214	100 lb = 12 gallon of 294 cyl in.
252.630 471 380 471	998.969 624 958	ISWS water: foot=7, pound=5.5, sec=15: 686/11 lb/ft³
252.631 578 947 368	998.974 004 578	Shuckburgh's water: 4800 gt = 19 cu in - 1919 rept.
252.651 025 526 335	999.050 901 643	Nines water = metric water / (15E9 G) G = 66.73E-12
252.704 281 068 689	999.261 488 548	60 french lb = 12.15-inch cube [Berriman]
252.720 000 000 000	999.323 645 480	A proposed equation from iwma.1821
252.732 155 002 407	999.371 709 667	49 lb / cylinder foot
252.745 862 798 450	999.425 914 022	0.216 grains / cu line, French measure
252.803 607 088 112	999.654 406 458	70 Paris lb / french cu foot
252.830 629 806 253	999.761 037 951	PA water using interim values. kg=18841, m=443.44
252.883 963 051 354	999.972 000 000	0.999972 metric water
252.884 216 126 750	999.973 000 728	1 kg/olitre, at 1000.027 cc
252.891 044 000 586	1 000.000 000 000	metric water = 1 kg/dm³
252.897 311 272 558	1 000.024 782 498	300 lb / 12.15-in cube [Berriman]
253.182 870 370 370	1 001.153 960 872	1000 oz = 1 cu foot
254.647 908 947 032	1 006.947 122 043	200 gt/cyl in

The Gallon

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This is a list of the various gallons, and their expressions in terms of the cubic inch.

Pre-Imperial Gallons

These were defined geometrically, either by their ratings in cubical inches, or in terms of a cylinder of defined height and diameter. A cylinder inch, is the measure, one inch in diameter and one inch high, and is used as a unit. The volume of cylinder gallons is thus D²H, in cylinder inches.

Year	Gallon
1490	The Winchester corn bushel,
1660	Ale and Beer Quart of 70½ cu in 282.
1695	Irish corn gallon of 272.25 cu in 272.5
1696	Corn bushel of 18½ d, 8 in h = 2738 cyl in 268.802 5
1706	Wine gallon - 6² × 7 in cylinder
1706	Wine gallon - 231 cubical inches 231

Of these, the corn bushel becomes the US dry bushel, and the wine gallon becomes the liquid gallon.

There is a heaped bushel, formed by adding a cone to the top of this, 19 1/2 inches in diameter and 6 inches high. This gives a total volume of 3498.5 cylinder inches, or 2747.7154746 cu inches, rounded legally to 2747.715 cu inches

The Winchester Wine gallon is derived from a definition switch. The conversions of wine gallons to the imperial gallon G1 is 231/277.274 or 0.83311093 imperial gallons, rounded to 0.8331. When the gallon was recalibrated in 1897, this definition was not updated, and the WWG is 0.8331 of 4.54609 Litre or 231.186020 cu in, or 3.78734759 litres. This is somewhat larger than the US gallon of 231 cu in or 3.7854117 litres.

The Imperial Gallon.

The imperial gallon is defined as the space occupied by that measure of water that would balance 10 pounds, when measured against brass weights at 62°F, and 30 inches Hg. Later, these conditions were defined in terms of their metric equalivants. The values shown here are those adopted for the W&MA of 1963.

Variable	1963 values	1824 values
Mass M	4535.923 g	70000 gr
Brass B	8316.000	8143
Water W	998.859	1000
Air A	1.217	1.2
Density D	998.859 g/L	252.724 gr/cu in
Gallon in cu in	4.545 978 64 591 48 L 4.546 091 878 49 cu dm; 277.420 994 598 uk cu in 277.420 275 561 ub cu in	277.273 915 097 cu in

The principle of bouyancy applies. Here, the balance is against apparant weights, which is the true weight reduced by the mass of the displaced air. This effectively reduces the density of the brass and water by that of air. The calculation goes like this. The mass m of brass occupies m/b units of volume. This volume has an apparent mass, as reduced by air of m÷b×(b-a). Likewise, the mass of water M has an apparant value of M÷w×(w/a). Now the apparent masses are equal, and because water is less dense than brass, displaces more air and so has a higher mass. Thus M = m÷b×(b-a)÷(w-a)×w. Now the volume is the space occupied by this mass at the density D, ie the gallon is M/D, or m×(b-a)×w÷b÷(w-a)÷D.

Putting the numbers into the above equations, we find: (here, 1 L = 1.000 028 cu dm)

• Gallon by 1824 values 277.273 915 097 cu in
• Gallon by 1963 values 4.545 978 64 591 48 L or 4.546 091 878 49 cu dm

The old gallon of 277.273 915 097 cu UV inches equates to 4.543 460 532 cu dm. The value 4.546 091 878 49 cu cm convers to cubic inches thus: by US, 277.417 883; by UI, 277.419 547; by UB, 277.420 271 656, by UL, 277.422 799 7; by UC, 277.427 014 3; by UV 277.434 583. Since at the time the value is set (1898), the current legal equalivants were the UC and as an auxhilary, UC, the value 277.420 can only arise from the UB.

The modern rating of the gallon, as defined in the act of 1985, is 4.546 09 litres. This rounds the equalivents of the gallon in terms of litre to five decimals, but in doing so, invalidates the 277.420 rating, since the gallon becomes 277.419 432 792 cu inches (UI). In terms of the other inches, it is by UB, 277.420 157, by US, 277.4177, by UK, 277.420 875 7, by UL, 277.422 685. We regard this gallon as an alternate definition of the previous.

The US gallons were the pre-imperial gallons, rounded to the nearest 0.001 cubic inches. From the abandonment of the colonies to the Mendenhall Order, the ultimate US standards were to replicate the UK ones, through legal replicas and conversion factors.
      From the time of the Mendenhall Order to 1959, the foot and pound were defined in terms of the international metric system, by US standard, and thence by the UI standard.

• Liquid Gallon = 231 cubic inches,
• Dry bushel = 2738 cylinder inches = 2150.420
• Proposed unified gallon of 270 cu in, and bushel of 2160 cu in.

The Gallon in in cubic inches

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The Gallon in Cubic Inches

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The gallon in cu inches 888888888888888	Description
224.000 000 000	The gallon at the Guildhall, as reported in Berriman [p161]. This is rated as 8 lb at 250 grains / cu in. This is an early rating of the Wine Gallon.
230.400 000 000	The Berriman version of the Wine Gallon, being 8 lb troy at 250 gr / cu in.
230.907 060 039	The wine gallon is described as a cylinder, 6 inches high and 7 inches diameter. This makes 294 cylinder inches.
230.996 969 400	This is 0.8331 G1, the original rating of the Winchester Wing Gallon. This is the legal value for coverting Wine Gallons to G1. It is carried across to G2.
231.000 000 000	A wine gallon, rated in cubic inches. The US uses this one.
231.118 155 15	This is the Winchester Wine Gallon, as 0.8331 G2, a version met in some conversion tables. This gallon appears in Horvath, for example. Note that 277.274 × 0.8331 = 230.996 969 4 cu in.
256.000 000 00	4 litres, or 8 metric lb of water, in metric inches
268.802 500 000	The US dry gallon is defined as an eighth of a bushel. The bushel rounds 2738 cylinder inches to 3 places of decimal.
268.802 521 423	A dry gallon, rated as 342.25 cylinder inches. The proportional bushel is described as a cylinder, 18½ in in diameter and 8 in high. This is the Winchester bushel of 1696-7.
269.391 570 045	A seven-inch cylinder. It should be remembered that the Japanese unit of capacity is defined as a cube, the side of which is 7/10 of their foot or 7/33 metre.
270.000 000 000	Proposed US gallon
272.000 000 000	In 1688 three gallon prototypes each yielded 272 cu in - Berriman pp158
272.160 000 000	Berriman rated this as the prototype corn gallon. This is 0.1575 cu ft, 8 such gallons make 63 lb (or 4½ stones) of wheat, taken at 50 lb per cu ft.
272.250 000 000	In 1695, the Irish parlement defined a corn gallon of this measure.
276.480 000 000	G0 The original intent of the imperial gallon was for water to be 1000 oz per cubic foot.
276.799 095 262	This is 10 lb of metric water, rated as 252.891 grains per cu in.
277.083 333 333	10 lb of Shuckburgh's water; that 1 troy oz = 1.9 cu in
277.202 568 400	A coal gallon, as in [MS], this is 33/32 × 268.8025.
277.228 285 714	[39.6 inch] cube = 224 gallons.
277.273 843 570	This gallon is 10 lb, where 1 cu in weighs 252.458 grains. This is the value used in [MS] and Gwilt.
277.273 915 097	The unrounded value of G1, expressed in UV inches. This is calculated from the following data: specific gravities of brass (8.143), water (1) and air (0.001 2), density of water at 62°F, 30 inHg, 252.724 gr / cu in, mass of brass 70,000 gr. This is 70000 ÷ 8.143 × 8.1418 ÷ 0.9988 × 1 ÷ 252.724.
277.274 000 000	G1: This gallon is the version as described in the act of 1820.
277.419 432 792	This is the UI gallon, rated as 4 546.09 cu cm, where the inch is 2.54 cms. The value of 4.546 09 is derived from 277.420 cu UB inches.
277.419 547 424	The gallon, as defined in 1963, here, the densities in g/L of brass = 8136, water = 998.859, air = 1.217, the litre = 1.000 028 cu dm, the conversions follow UI. The gallon becomes 4.545 964 591 48 litres, or 4.546 091 878 49 cu in.
277.419 994 055	This is 10 lb, where 1 cu in corresponds to 252.325 grains.
277.420 000 000	G2: This is the normal form the new gallon is quoted in.
277.420 184 147	The gallon as recalculated in 1898, where brass = 8143, air = 1.218 738, water = 998.8611, m = 4535.9243, gives 1 gallon as 4.545 963 157 44 cu dm. This is converted using the UB factors.
278.400 000 000	four Quarts, measured from the 1758 measuring of a qt dated 1601. This appears to be the origion of the imperial gallon [a quart that gave nearly 20 oz of water].
280.000 000 000	Berriman's proposed rating of this gallon (276-282). This is 10 lb at 250 gr / cu in.
282.000 000 000	This gallon is defined in terms of a Quart, rated as 70½ cubic inches. This is based on the 1688 measuring of the 1601 quart.
288.000 000 000	1/6 cu foot; 4.5 L in metric inches.
300.000 000 000	5L, or 10 lbm of water, in metric inches

The values of Pi

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This section arranges in increasing order, the various appoximations to the ratio of the circumfenence to the diameter of the circle, expressed normally by the greek letter pi π.

Pi is less fundemental than e, for example. Radians on the other hand, arise quite naturally from the calculus (which is why they're used). While there should exist a constant k that nk radians make the circle, the is no over-riding reason why n should be two, in fact there are sound arguements to support 1, 2, 4, 6 and 8.

The values of Pi

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Pi in decimals 888888888888888	Description
3.000 000 000 00	3 This is the value of pi, assuming the hexagon represents the circle.
3.072 000 000 00	A five inch sphere has the same volume as a four inch cube.
3.125 000 000 00	This value of pi appears as one of the forms in Berriman. It was also known as an approximation to the Egyptians.
3.136 000 000 00	A five-inch sphere has a volume of 65 1/3 cu in (= 2744 / 42).
3.141 361 256 54	600/191. This is the value of pi, when one radian equals 57.3 degrees. As an expression of 1/pi = &38&24/120, is accidently equal to the solid angle of the 120 cell /5,3,3, this being 38s 24f or &38&24/120 of the 4 sphere.
3.141 567 230 22	823543/262144: This appears to be the best approximation to pi, using numbers whose prime factors are 2, 3, 5 and 7. The value is 7**7/8**6. Written in octal, the value of pi is 3.1103 7552 52, and this number is 3.1103 67.
3.141 592 653 59	Pi to 11 decimals. (twe: 3:16E8.E212.7796)
3.141 592 920 35	355/113, a very good approximation to pi. If ye want to run an experiment, the result depends on pi, ye can fake the results using this approximation.
3.141 600 000 00	3.1416: This value of pi is known in India. Interistingly, the recripical of this written in twelfty is 0:38.23.78.23.78.23.78... Note: 3.1416 is 187×168.
3.141 640 786 50	1.2 phi², this is known from a number of different sources.
3.141 666 666 66	377/120. This version of Pi is known in the sexagesimal form 3&8&30/60. In the form 3&17/120, it reflects a version of 1.2 × phi². In this form, it can be written as 1.2×377/144, where 377 and 144 are fibanacci mumbers.
3.141 818 181 81	864/275. This equates a cubic foot with 2200 cylinder inches.
3.142 337 619 40	24/35 sqrt(21). This comes from 1 cyl ft = 2800 hoppus in = 36 Litre.
3.142 561 983 47	1521/484: This is 3&3&3/22. This ratio is a square (of 39/22), and thus a 44-inch circle has the same area as a 39-inch square.
3.142 696 805 27	20/9 sqrt(2): This is the square root of 800/81. This is the value of pi, implied by equating the quadrant to 10 parts, where the inscribed square has an edge of 9. This is known to the ancient Egyptians. This value also makes an f-unit equal to a cubic inch on the surface of an eighteen-inch four-sphere, there being 14400 f-units making full space.
3.142 857 142 85	22/7: 3&1/7. This is the most common approximation to pi, used in calculations. Metrologically, it is not at all common. The one place where it arises is the rating of the Wine Gallon as 294 cylinder inches, and 231 cubical inches.
3.146 108 329 53	6912/2197: This equates a 13 inch cylinder with a 12 inch cube.
3.150 000 000 00	63/20: This value of pi appears in Berriman, etc, as a form of the geographic pi.       The Swedish army mill uses this pi.
3.152 161 253 53	The geographic pi , defined as length of equator ÷ polar axis. Both of these measures have been extensively used in metrology.
3.160 493 827 16	256/81, a value known to the Egyptians. A nine-inch circle has the same area as an eight-inch square.
3.162 277 660 17	sqrt(10): At one time, the square root of 10 was put forward as a version of pi.
3.200 000 000 00	32/10: The value pi used in the NATO army mil.
3.210 879 629 63	277.42/86.4, A cylinder foot is 5 gallons (BHP Lysaght)

army mil

The circle is divided into an exact number of units, which are meant to correspond to milliradians. This generates a pi-approximate, eg 3.15 or 3.2

Cylinder measure

The cylinder of unit height, and unit volume.

Geograpic pi

The ratio of the polar axis to the circumference of the Earth. Both measures have been used extensively in metrology.