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Standard Wheat

Various Metrological Wheat Units
grain/in³
888888888888888
g/dm³
888888888888888
Description
187.012 987 739.500 237 1 wine gallon = 8 tower lb wheat.
189.243 746 748.321 265 1 imperial bushel = 60 lb avoirdepoise
192.857 142 759.220 243 224 cu in = 8 tower pounds.
194.726 070 770.000 000 1 hectolitre = 77 kilograms (finland)
195.087 342 771.428 571 27/35; = tower / avoirdepoise.
195.310 684 772.311 589 1 us bushel = 60 lb avoirdepoise
200.000 000 790.854 420 216 cu in = 8 tower pounds
201.860 000 798.209 366 1 imperial gallon = 8 lb avoirdepoise.
Wheat in tower pounds is the same as water in avoirdepois lb
various metrological coal units
grain/in³
888888888888888
g/dm³
888888888888888
Description
235.688 043 651 931.974 655 657 112 lb = 12 coal gallons = 3326.32 cu in

Standard Water

Tables: Home - m2in - kg2oz - gm2gr - Water - Gallon - Gal2in - Inches - Technic - Pi
Various Metrological Water Units
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

Tables: Home - m2in - kg2oz - gm2gr - Water - Gallon - Gal2in - Inches - Technic - Pi

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)

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.

The Gallon in in cubic inches

Tables: Home - m2in - kg2oz - gm2gr - Water - Gallon - Gal2in - Inches - Technic - Pi

The Gallon in Cubic Inches
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

Tables: Home - m2in - kg2oz - gm2gr - Water - Gallon - Gal2in - Inches - Technic - Pi

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
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.


© 2003-2004 Wendy Krieger