How to Find the Height of Heaven
How the Chinese discovered the Pythagorean Theorem hundreds of years before Pythagoras
Discoveries are often made by multiple people, but we tend to only celebrate one of those folks.
This is just as true for the theory of evolution (Wallace, but also Darwin), as it is for lightning rods (Diviš, but also Franklin), as it is for the electrical telegraph — invented by Wheatstone but also Morse, whose first message is a favorite of trivianauts and bibliophiles alike (“what hath god wrought”, Numbers 23:23).
The concept of multiple discovery also happens to apply to a favorite hobbyhorse of mine, the Pythagorean Theorem — i.e., that natty formula that allows you to deduce the area of a square alongside a triangle’s hypotenuse, which in turn allows you to develop trigonometry, which in turn allows you to (eventually) navigate the ocean using a sextant and (even further eventually) calculate things like the height of Mount Everest without climbing it and the fastest route via Google Maps.
Now any Tom, Dick, or Platonist can tell you that the Pythagorean Theorem was first set down as an axiomatic proof in The Elements around 300BCE. It is slightly less well known (tho very wikipediable) that parts of the theorem were known as early as 3000 BCE as far away as India and China. As with all multiple discoveries, the interesting question is…how? How did multiple people discover the same thing? And in this case, given that the theorem, y’know, empirically describes geometric reality, was there any similarity in the thought process that led people thousands of miles distant and thousands of years apart to discover it? I mean, everyone was just trying to create, like, wagon wheels…right?
Hoo boy, not at all! The triangles in Babylon were the triangles in Punjab were the triangles in Zhou, but the reason for developing those triangles was wildly different. Those differences reveal quite a lot about how different cultures pursued knowledge, and even how they valued the practice of learning. My favorite example comes by way of China.
Thirteen centuries before Euclid put reed to papyrus, around the year 1600 BCE — just about the time the Egyptians were learning how to leaven bread along the Nile and the last wooly mammoth was laying down to die in Siberia — the Shang Dynasty came to power. This was in the Yellow River valley not too far from where Beijing is today, about 400 kilometers west as the three-legged crow flies.
The Shangs fought with axes made from bronze. They wrote on oracle bones. Their king-priests divined the will of ancestors by cracking open turtle shells, sacrificed humans to a sky god, and dismembered bodies for display — mostly, it seems, while shitfaced from wine. They ruled for five hundred years. Their reign came to an end, as the story goes, when one particularly depraved Shang king made a certain count eat a meat pie of his own son. That count, none too pleased, stoked a rebellion. His remaining sons overthrew the depraved king, who committed suicide by lighting himself on fire. That was the end of the Shang. Then came the Zhou who, as we’ll see, developed a certain fascination with circles, squares, and triangles.
To justify their takeover of Shang wealth and territories, the Zhou claimed something that, thousands of years later, will ring familiar to politicians, priests, and twitter zealots alike: moral superiority.
The Shang, the Zhou argued, were drunks. Furthermore, the Shang’s rule by royal lineage had led to despotism (see, e.g., the aforementioned meat pies). The Zhou rulers, by contrast, supported a morally correct transfer of power. They called this power the mandate of heaven.
The mandate went like this:
- Heaven grants the emperor the right to rule
- The emperor’s virtue determines his right, and
- An emperor can lose that right if he, seemingly or otherwise, loses his virtue.
- Anybody can rule, not just royalty.
This was a BHD (Big Historical Deal).
By doing away with rule by royalty, the mandate opened leadership to the people and justified rule by rebellion and force (“I have the mandate, follow me!”). By tying leadership to virtue, the mandate also required rulers to be responsible (“I treat you well, keep following me!”). But by leaving “virtue” open to interpretation, the mandate incentivized rulers to, well, make shit up. In practice, it meant that rulers had to secure the mandate through tangible and explicable communications with heaven, and explain every chance event (comets, earthquakes, famine, etc) as a positive sign of heaven’s unwavering aegis.
As you might imagine, trying to convince people at every moment that heaven loves you can be ≋c≋r≋a≋z≋y≋ ≋m≋a≋k≋i≋n≋g≋. Rulers had to make sacrifices on mountain tops and create calendars to predict celestial events and build temples to receive heaven’s blessing. In other words, they had to interpret and enact the reality they wished their followers to believe in (you might say they were…ZhouAnon 😬😬😬 ). But it’s there, in the midst of all that performative interpretation, that the triangles come in.
The Zhou believed Earth was a square and heaven was a circle. As in: the Earth was a flat, four-cornered thing, and heaven existed as a flat disc directly above it. The sun, according to this conception, was 80,000 li above (roughly 20,790 miles):
To make this concept more accessible, they likened the square to a chariot and the circle to the canopy that stood above it. They called this concept “Canopy Heaven”.
Over time, the idea of Canopy Heaven fell out of favor. Some argued that if Heaven were exactly round and Earth exactly square, and both were exactly the same size, then that meant that the four angles of square Earth would not be well-covered by circular heaven.
Owing to this lack of figurative accuracy, the concept of Canopy Heaven began to be replaced by another concept — Spherical Heaven — whose main assumption was that Heaven contains Earth in the same way that an egg contains the yolk. Clearly, more rigorous minds had prevailed.
But some mathematicians couldn’t let the idea of Canopy Heaven go. The author of the Zhou Gnomon, a mathematical classic compiled in the first century BCE, turned the problem of joining a round Heaven and a square Earth into the problem or arranging a circle and a square on a plane, especially one within the other. From that attempt to reason out the physical connection between the circle of Heaven and the square of Earth, some historians argue, the author deduced an instance of the Pythagorean Theorem.
So to sum up: The ancient Chinese derived a geometric truth because they believed they had to keep despots off the throne, and in order to keep despots off the throne they had to maintain heaven’s favor, and in order to maintain heaven’s favor they had to prove the earth was a square and heaven a circle. Which is to say that even the most enthusiastically wrongheaded folks can, in the attempt to justify their own moral superiority, make some useful discoveries — it’s just a long way to go for a ham sandwich.
That would be the end of our story but, to bring this argument to a π-like full circle, it’s important to note a few important contextual bits and bobs.
While the ancient Chinese did develop geometric principles, and they did develop those principles in pursuit of judging heaven’s height, there is debate about whether their mathematicians used “proofs”. This may seem like a minor point, but the debate suggests that Chinese mathematicians couldn’t “discover” the Pythagorean Theorem if they didn’t “show their work” in the same way the Greeks did.
But that equivalency suggests a biased reasoning. Perhaps looking for a Chinese equivalent to Grecian methodology is, itself, an ethnocentric expectation. As some historians have noted, giving a Euclid-like proof wasn’t as important to Chinese mathematicians as explaining the use of the methods they were expounding to solve certain problems. It’s only the Western value system, which was written into the history books during the 18th century’s so-called opening of China, that argues for the supremacy of the Grecian way of thinking.
This debate about this aspect of ancient geometry extends like a straight line into infinity, and it’s beyond the scope of this piece to address it all. Suffice that, for our purposes, it’s enough to say that the Pythagorean Theorem was discovered, in one way or another, and for one reason or another, several times by several different people on on at least two separate continents.
Some of those people were searching for the heights of heaven.
Others were searching for the heights of philosophy.
There’s an elegant equivalence there, if you do the math.
Regardless, no matter where you are, like Euclid said: All right angles are equal to each other.
Sources and Further Reading
- Karine Chemla, The History of Mathematical Proof in Ancient Traditions, (Cambridge University Press, 2012)
- Dirk L. Couprie, When the Earth was Flat: Studies in Ancient Greek and Chinese Cosmology, (Springer Nature Switzerland, 2018)
- Christopher Cullen, Astronomy and Mathematics in Ancient China (New York: Cambridge University Press, 2007)
- Michael Loewe, Edward L. Shaughnessy, The Cambridge History of Ancient China: From the Origins of Civilization to 221 BC (Cambridge University Press, 1999)
- Lillian Lan-ying Tseng, Picturing Heaven in Early China (New York: Harvard University Asia Center, 2011)