Example 3: If we go farther north, say to a calibration latitude 4/7ths of the way from the equator to the pole, a sacred latitude to the Egyptians, we will be setting up our pendulum near the Neolithic site of Avebury in England at 51°25' or even Greenwich in London at 51°28' In this case our sidereal-second foot is 0.247092 meters and our reference latitude is to the south at 48°30\ Probably there was such a pendulum station at this latitude in England because the English foot can be derived from this foot, and, for very good reasons, the standard reference latitude of the British Navy was 48°30'.
Circles
The ancients used different circle templates depending upon which estimate of pi they employed.
(a) Using 3+1/6 = 3.16667 for pi and a diameter of 24 we get a circumference of 76. This is the template used for the English foot. Eighty times 76 equals 6080 (8 stadia of 760 feet) which is the number of English feet in a minute of arc, which gives us 131,328,000 English feet in the circumference at the reference latitude of 48°30'.
(b) Using 3+1/8 = 3.125 for pi and a diameter of 24 we get a circumference of 75. This is the template used for the Nautical foot. Eighty times 75 equals 6000 (10 stadia of 600 feet) which is the number of nautical feet in a minute of arc, which gives us 129,600,000 Nautical feet in the circumference at the specified reference latitude.
The Roman foot also uses this template: 75x1,800,000 = 135 million, since it is based on a diameter of 43,200,000 feet at its reference latitude.
(c) Using 3+1/7 = 22/7 for pi and a diameter of 21 we get a circumference of 66. This is the template used for the foot which divides the minute of arc into 5280 feet, since 80 times 66 = 5280 (8 stadia of 660 feet). This foot is exactly 76/66 times the English foot in length and 75/66 times the Nautical foot in length. The value of 5280 English feet in a mile is probably an error introduced into the English metric system by an error in the translation of Ptolemy’s writings from the Arabic which led the unsophisticated Europeans to believe that their planet was smaller than it really was. Early eighteenth century English texts such as Cocker's Arith-metick still assumed that there were 5280 English feet in the minute of arc because of this transcription error.
The Circumference of the Earth
Aristotle’s mathematikoi quoted 400,000 stadia as the circumference of the Earth. This is a stadium of 300 feet of a very common Mesopotamian foot, computed as the 60th part of a nine sidereal-second pendulum that divides the Earth’s circumference into 120 million parts. It is 27/20ths of the standard sidereal-second pendulum calibrated at the 30th degree of latitude, a foot of 0.33294 meters, and was the commonest foot used in Greece at the time according to Stecchini.
Archimedes quoted 300,000 stadia for the circumference. This is a stadium of 300 Roman cubits. The Roman cubit divides the Earth’s circumference into 90 million parts and is 9/5ths of the standard sidereal-second pendulum, 0.44392 meters. It is the 20th part of a six sidereal-
second pendulum. The Roman foot is the 30th part of the same pendulum.
Eratosthenes never had to leave his library. All he had to do was dust off a scroll that Alexander had looted from an Egyptian school. He didn’t even bother to get the latitude of Alexandria right. His quote of 250,000 stadia in the Earth’s circumference is a reference to a stadium of 300 cubits of the standard Great Cubit which divides the Earth’s circumference into 75 million parts. It is the 24th part of a pendulum that beats out 12,000 cycles per sidereal day, 2.16 times the sidereal-second length, 0.53270 meters, and was the standard cubit of Egypt in his day.
The Arabic or Black Cubit, which dates to a time much earlier than the Moslem conquests, divides Earth’s circumference into 81 million cubits, with 3750 to the minute of arc. At 0.49324 meters it is exactly twice the length of the one sidereal-second foot and was probably computed as the eighth part of a four sidereal-second pendulum.
The Nautical foot which divides the Earth’s circumference into 129,600,000 parts, 6000 to the minutes of arc, 100 to the second of arc, is 5/4ths of the one sidereal-second foot. When measured at the sacred latitude of 51°25’43” it is exactly 76/75ths of the English foot, the same ratio used by the British Admiralty.
The Egyptian Royal Cubit...
—End of fragment