From Chiggala Sutta SN56.48
“Monks, suppose that this great earth were totally covered with water, and a man were to toss a yoke with a single hole there. A wind from the east would push it west, a wind from the west would push it east. A wind from the north would push it south, a wind from the south would push it north. And suppose a blind sea-turtle were there. It would come to the surface once every one hundred years. Now what do you think: would that blind sea-turtle, coming to the surface once every one hundred years, stick his neck into the yoke with a single hole?”
“It would be a sheer coincidence, lord, that the blind sea-turtle, coming to the surface once every one hundred years, would stick his neck into the yoke with a single hole.”
"It’s likewise a sheer coincidence that one obtains the human state. It’s likewise a sheer coincidence that a Tathagata, worthy & rightly self-awakened, arises in the world. It’s likewise a sheer coincidence that a doctrine & discipline expounded by a Tathagata appears in the world. Now, this human state has been obtained. A Tathagata, worthy & rightly self-awakened, has arisen in the world. A doctrine & discipline expounded by a Tathagata appears in the world.
“Therefore your duty is the contemplation, ‘This is stress… This is the origination of stress… This is the cessation of stress.’ Your duty is the contemplation, ‘This is the path of practice leading to the cessation of stress.’”
Although it’s repeatedly said that human life is very rare with minuscule probability, I want to calculate the probability.
According to source[1], bow width ranges from 4inch - 12 inches. As an average 8 inches is taken to be the bow width(hole is assumed to be a circle - actual holes are not).
Area of the yoke hole:
- For a yoke hole with a diameter of 0.2032 meters, the radius r is 0.10 meters.
- Therefore, Ayoke=π×(0.10)^2=π×0.001≈0.03141 m^2
Surface area of the Earth[2]
- Aearth ≈ 5.1 x 10^8 km2
Ratio:
R=Ayoke / Aearth= 0.03141 / 5.1E14 = 6.15882352941 x 10^-17
Probability = P =6.158 x 10^-17 x 100 = =6.158 x 10^-15 %
Given that turtle rises to surface every 100 years,
- Expected number of attempts to try = E= 1/P
- E= 1 / (6.158 * (10 ^ (-15)) = 1.6239e+14
- E×100 =1.6239e+14×100=1.6239 x 10^16 years
1.6239 x 10^16 years, or 16.2 quadrillion years, for the turtle to encounter the yoke hole.
Ratio of humans to animals
- Number of insects = 10 quintillion (10^19) (estimated)[3]
- mammals = 130 billions[4]
- birds = 50 - 430 billion[5]
- fish = 3.5 trillions[6]
Total number of animals = 10 quintillion (10^19) (from insects)
Total number of humans = 8.2 billion[7]
Ratio of human to animals = 8.2E9 / 10^19 = 8.2 x 10^-10}
meaning that for every human, there are about 12.2 billion animals.
Kalpa - eon
From Pabbata Sutta( SN 15:5)
Suppose there were a great mountain of rock—a league(yojana) long, a league(yojana) wide, a league(yojana) high, uncracked, uncavitied, a single mass—and a man would come along once every hundred years and rub it once with a Kāsi cloth. More quickly would that great mountain of rock waste away and be consumed by that effort, but not the eon. That’s how long, monk, an eon is. And of eons of such length, not just one eon has been wandered-through, not just one hundred eons have been wandered-through, not just one thousand eons have been wandered-through, not just one hundred-thousand eons have been wandered-through.
From Wikipedia[8]
yojana:= a measurement of very uncertain length, but sometimes taken to be about 4,000 times the height of a man, and so approximately 4.54 miles (7.31 km).
Volume of the mountain = 7.31 km ^ 3 = 390.617 cubic kilometers.
Suppose fine cloth is 1m x 1m(Area) and dust wipes away 1mm(height),
Dust wiped by volume = 1 x 1 x 0.001 = 0.001 m^3
Ratio = 390.617×10^9 m^3 / 0.001 m^3=390.617×10^9×10^3=390.617×10^12
100 years per wipe,
Total years = 390.617×10^14 years = 3.91x10^16 years (39.2 quadrillion years)
From SN 15:6
“It is, monk,” said the Blessed One. “Suppose there were an iron fortress—a league long, a league wide, a league high—full of mustard seeds packed tight, and a man would come along once every hundred years and take from it a single mustard seed. More quickly would that great heap of mustard seed waste away and be consumed by that effort, but not the eon. That’s how long, monk, an eon is.
Volume of fort(equal to mountain) = 391.911×10^9 m^3
- Volume of mustard seed:
- For a mustard seed with a diameter of 2 millimeters (0.002 meters)[9],
- V=4/3x π(0.001)3 = 4/3xπ(1×10−9)=4.19×10−9 cubic meters[10]
Ratio = volume of fort / volume of seed = 9.35348448687 x 10^19
Number of seeds = 9.35348448687 x 10^19 = 9.353 x 10 ^19
Given a seed is removed every 100 years,
Total years = 9.353 x10^19 x 100 = 9.353 x 10^21 years
Sources
[1]: Building an Ox Yoke. Tillers International, n.d. https://plainfieldmahistory.org/wp-content/uploads/2020/10/BuildinganOxYokeTechGuide.pdf.
[2] Williams, Matt. “What Is the Surface Area of the Earth? - Universe Today.” Universe Today, February 13, 2017. What is the Surface Area of the Earth? - Universe Today.
[3] “Lists of Organisms by Population.” Wikipedia, December 20, 2024. Lists of organisms by population - Wikipedia.
[4] “Lists of Mammals by Population.” Wikipedia, August 13, 2024. Lists of mammals by population - Wikipedia.
[5] Samurovic, Katarina. “How Many Birds Are in the World? In Numbers & Biggest Dangers.” Birdzilla - Enjoyin’ Birds, October 7, 2024. How Many Birds Are In The World? In Numbers & Biggest Dangers.
[6] Chepkemoi, Joyce. “How Many Fish Live in the Ocean?” WorldAtlas, April 25, 2017. How Many Fish Live in the Ocean? - WorldAtlas.
[7] “World Population Clock: 8.2 Billion People (LIVE, 2024) - Worldometer,” n.d. World Population Clock: 8.2 Billion People (LIVE, 2024) - Worldometer.
[8] Wikipedia contributors. “Buddhist Cosmology.” Wikipedia, December 18, 2024. Buddhist cosmology - Wikipedia.
[9]. “Mustard Seed.” Wikipedia, December 4, 2024. Mustard seed - Wikipedia.
[10] Pamuła, Hanna, Piotr Małek, and Mateusz Mucha. “Volume Calculator.” Omni Calculator, June 12, 2024. Volume Calculator.