More infor on Yojanas by HG Sadaputa Prabhu
Exact Science In the Srimad-Bhagavatam | Krishna.com
Exact Science In the Srimad-Bhagavatam | Krishna.com
Complexity: Medium by Sadaputa Dasa A unit of measure known as the yojana hints at advanced astronomical knowledge in the ancient Vedic civiliz…
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Exact Science In the Srimad-Bhagavatam
by Sadaputa Dasa
A unit of measure known as the yojana hints at advanced astronomical knowledge in the ancient Vedic civilization.
An encyclopedia article states that in early times length was defined by the breadth of the palm or hand, and the length from the elbow to the tip of the middle finger (the cubit). The article goes on to say, “Such standards were both changeable and perishable, and only within modern times have definite unchanging standards of measurement been adopted.” (Microsoft Encarta)
The Middle Ages certainly saw many conflicting and poorly defined standards of weights and measures. But exact standards of measurement are not solely a modern invention.
Consider this example. In tenth-century England, King Athelstan decreed that the king’s girth, in which the king’s peace is in force, should extend from the royal residence for a distance of 3 miles, 3 furlongs, 9 acres, 9 feet, 9 palms, and 9 barleycorns. This sounds quaint. But it defines a circle with a diameter of 36,500 feet – almost exactly 1/10 of a degree of latitude in southern England.
Measuring with Latitude
To define a unit of length exactly, it is natural to use latitude as a standard, because latitude derives from the size of the earth, a constant that can be measured astronomically. So if a fire or invasion destroys the standard measuring rod stored in some government building, astronomical readings can be used to restore the lost standard. Of course, it seems unlikely that accurate astronomical measurements were being made in England in the days of King Athelstan. But if we look into the history of weights and measures, we find that distances were gauged in terms of latitude in ancient times, and medieval societies inherited many exact standards of measurement. These included volumes defined as length cubed and weights defined by filling such a volume with water.
The Greek astronomer Eratosthenes is usually credited with being the first to measure the size of the earth by observing latitudes (see Figure 1). He is said to have noted that the sun, when directly overhead at Syene at the Tropic of Cancer, casts a shadow of 7.2 degrees at Alexandria. Knowing the distance between Syene and Alexandria, he could compute the length of a degree of latitude and estimate the circumference of the earth.
But there is reason to believe that the size of the earth was known long before Eratosthenes. The Italian scholar Livio Stecchini has given extensive evidence that the ancient Egyptians laid out their country using latitude and longitude. He argues that they had accurate knowledge of the dimensions of the earth and that such knowledge was inherent in the design of the great pyramid at Giza. Since the great pyramid dates to about 2500 B.C., this implies that the earth was measured scientifically at least that long ago.
Defining the Yojana
Turning to India, we find a unit of distance, called the yojana, that at first glance seems as ill defined as the medieval English furlong or foot. The yojana is defined to be either 16,000 or 32,000 hastas, where a hasta, or cubit, is 24 angulas, or fingers. That there were at least two sizes for the yojana is upheld by the writings of classical Indian astronomers. The fifth-century astronomer Aryabhata used a yojana of about 8 miles, and the astronomy text Surya-siddhanta a yojana of roughly 5 miles.
The first hint of the ancient history of the yojana comes from Strabo, who describes the experiences of Megasthenes, a Greek ambassador to India in the period following Alexander the Great. Strabo cites Megasthenes as saying that along the royal road to the Indian capital of Palibothra (thought to be modern Patna), pillars were set up every 10 stadia (see Figure 2). The British scholar Alexander Cunningham argues that the pillars marked an interval of one krosa. Since there are traditionally 4 krosas per yojana, this implies 40 stadia per yojana.Stecchini gives 400 cubits per stadium, and this implies 16,000 cubits per yojana.
Since the smaller of the two definitions for the yojana assigns it 16,000 hastas, we can tentatively identify the hasta, or Indian cubit, with the Greek cubit. This unit is well known, and it enables us to compute the length of the yojana. The Greek cubit is 462.42 millimeters. This gives us a small yojana of about 4.6 miles, in rough agreement with texts such as the Surya-siddhanta.
Stecchini points out that the stadium was defined as 1/600 of a degree of latitude. This would mean that there are 15 small yojanas per degree. Likewise, there are 60 krosas per degree, or 1 krosa per minute.
Here we must make a technical observation about latitudes. Consider the earth to be a sphere, rotating on a line through the north and south poles called the polar axis. The latitude of a person facing north at some point in the northern hemisphere is the angle from his horizon up to the polar axis (see Figure 3). That angle is 0 degrees at the equator and grows to 90 degrees at the North Pole. The length of a degree of latitude is the distance a person would have to travel north for his latitude to increase by 1 degree. On a perfect sphere, this distance would be the same at all latitudes. But the earth is slightly flat at the poles and bulges at the equator. This makes for a degree of latitude slightly smaller at the equator than further north (see Figure 4).
Stecchini noted that the Greek stadium is 1/600 of a degree of latitude at Mycenae in Greece, and he argued that it was deliberately defined this way in ancient times. I propose that to define the yojana in India the degree of latitude at the equator was used. This means that the hasta should be 460.7 millimeters instead of 462.4 millimeters (and the yojana would still be about 4.6 miles). I shall point out below why this fine distinction is important.
At first glance, the yojana of 32,000 hastas should be twice as long as this, or about 9.2 miles. But there is reason to think that these two yojanas use different standards for the hasta (see Figures 5 and 6).
Hiuen Thsang, a Buddhist pilgrim who visited India in the seventh century, wrote of yojanas in terms of a Chinese unit of measure called the li. He reported that a yojana consisted of 40 li according to Indian tradition but the measure in customary use equaled 30 li and the measure given in sacred texts was only 16. The li has taken on many values during China’s history. But using values for the Thang dynasty, when Hiuen Thsang lived, we can compute that the yojana of 16 li matches the small yojana of 4.6 miles.
Could the yojana of 30 li match the larger yojana of 32,000 hastas? If it does, then the larger yojana has to use a slightly smaller hasta, 30/32 as long as the hasta in the shorter yojana. Multiplying our hasta of 460.7 millimeters by 30/32, we get a smaller hasta of 431.9 millimeters. The larger yojana of 32,000 hastas then comes to 8.59 miles. At the equator, that is 1/8 of a degree of latitude.
In an investigation to be reported in a later article, I found that the geocentric orbits of the planets Mercury, Venus, Mars, Jupiter, and Saturn align closely with the dimensions of dvipas in Bhumandala. Bhumandala and dvipas are features of cosmic geography defined in the Fifth Canto of the Srimad-Bhagavatam. To align planetary orbits with dvipas we need to be able to convert the yojanas used in the Bhagavatam into the miles or kilometers of modern astronomy. I found that the alignment of orbits and dvipas works well if we assume about 8-1/8 miles per yojana.
To compare orbits with the structure of Bhumandala, I used modern ephemeris programs for orbital calculations. I was most