Actually, the Egyptian writing system is a fascinating combination of both pictographic and alphabetic writing systems. It is also far easier to read than most people imagine. To really appreciate Ancient Egyptian, we have to understand how the Ancient Egyptians wrote their language. To do that, let’s look at some dirty chain texts first.

A friend of mine recently sent me an emoji-filled holiday chain letter. This is an entire genre of spam texts. In case you haven’t been exposed to these wonderful messages, here are some examples:

Generally, these emoji-filled exhortations are text messages, usually wishing you a happy holiday, promising you sex if you forward it to another person and cursing you with a lack of sex if it isn’t forwarded. These messages are usually littered with ham-fisted sex puns.

Emoji chain texts also happen to be the most perfect modern analogy for the writing system of Ancient Egypt that I’ve ever encountered.

The Rebus Principle: a literary equation

Let’s consider an example. From the first chain text:

Consider the function of the “4” emoji. This is nominally a pictorial representation of the number “four,” using the Arabic numeral system. However, in American English, the pronunciation of “four” coincides with the pronunciation of the preposition “for.” Hence, the pictogram “4” can be used to mean “for.” This is an example of the rebus principle, in which words are represented by pictograms that sound the same. Here’s another of the rebus principle, from the Egypt Exploration Society’s webpage:

A picture of a bee followed by a picture of a leaf would be pronounced “bee-leaf,” a homophone of the word “belief.” Thus, the bee and the leaf symbols, together, represent the totally unrelated concept “belief.”

The Ancient Egyptian writing system is based on the rebus principle. Originally, the “mouth” hieroglyph represented the concept of “mouth,” and was pronounced something like r.

Not long after the invention of Egyptian writing, the mouth glyph was assigned the phonetic value of r. A set of these signs was standardized, creating the hieroglyphic alphabet. Here’s the (Middle) Egyptian alphabet:

These signs are used to spell out the sounds of Egyptian in the same way that the Roman alphabet is used to spell the sounds of English. Mostly. In Egyptian, like a lot of Semitic languages (Hebrew, Arabic, etc.), vowels tend not to be explicitly written out. Only the consonants were written down, along with pseudo-vowels like i, sometimes called a “weak consonant.” This lack of vowels in writing leads to a lot of homophones in Egyptian, words that sound (or at least are written) the same but have different meanings.

Semantic determinatives as seen through Earth emojis

Let’s now consider a second example, from the second chain text:

Here, consider how the emojis following the word augment its meaning. It begins with “Happy Earth Day,” followed by an emoji of a plant and three of the Earth. The compound noun “Earth Day” is composed of two words written in the Roman alphabet. The individual characters (a, p, y, etc.) tell the reader how the words are pronounced. This is the hallmark of an alphabetic system; an individual character d represents the sound of a single consonant, and multiple characters representing distinct sounds, like d a y, are placed in sequence to form a word with lexical meaning, “day.” The characters tell you how the word is pronounced, and collectively form a written representation of both the concept “day” and the sound “day.”

The sound-signs forming the word “Earth Day” are followed by a picture of a plant and three pictures of the planet Earth, indicating that “Earth Day” is a concept associated with living, growing things and the planet Earth. In other words, the pictograms following the alphabetic characters add shades of meaning to the phrase “Earth Day,” clarifying the category of concept to which this word belongs.

This is precisely how Egyptian words are formed. Paraphrasing from James Middle Egyptian Grammar, Egyptian words are commonly spelled out alphabetically, but also followed by an additional sense-sign, called a determinative, that adds context and meaning to the sound-signs.

For example, the word ra is written as:

It consists of two alphabetic signs, the mouth hieroglyph, pronounced r, and the arm hieroglyph a, pronounced something like the Arabic ayin. The word ra is followed by this circular determinative, which indicates the meaning of the word ra.

Can you guess what ra means from the determinative sign? You probably can: ra means “sun,” and the determinative is a picture of the sun. The image of the sun clarifies the meaning of the sound-glyphs r and a.

Disambiguation by means of eggplants and seated gods

Let’s consider a third illustrative example. From the third chain text:

The emojis clarify the meaning of “Hot Dog.” The compound noun “Hot Dog” is followed by a peach and eggplant emoji, commonly used to mean “butt” and “penis” respectively. Here, the eggplant and peach emojis serve an important semantic function — they clarify the ambiguity in the sentence “I Want To Eat Your Hot Dog” by explicitly informing the reader, using the eggplant determinative, that “Hot Dog” is a euphemism for “penis.” Thus, instead of the sentence indicating a desire to eat a delicious, all-beef frankfurter, it indicates a desire to perform oral sex.

Ancient Egyptian uses determinatives in exactly the same way as the chain text uses the peach and the eggplant. Returning to our example of ra, consider these two examples of Egyptian words, both spelled ra:

The first word is followed by the “sun” determinative, and thus refers to the concept of the sun, i.e., the ball of fire in the sky. The second is followed by the seated god determinative, and instead of referring to the sun itself, it refers to the sun god Ra. The determinative serves to clarify which concept, both spelled ra, is being referred to in the text.

The determinative is extremely important to understanding written Egyptian, due to the number of homophones in the written language.

Lesson 4: Illustrative Examples

We also notice that in these emoji chain texts, the short, common words without really concrete meanings (like “is” or “to”) are not followed by emoji determinatives, whereas nouns like “Patriotic Daddies” and “COCKtober” are followed by one or two determinatives indicating their meaning or associations in the context of the sentence. 

Egyptian follows the same pattern. Short, common words, like m, meaning “in” or “with,” are composed of alphabetic signs alone, without determinatives.

However, nouns and verbs usually consist of a series of alphabetic signs that indicate the pronunciation of the word, followed by a semantic determinative that indicates its sense, category, or associations.

Let’s consider the example of the Egyptian verb “beget,” meaning “to bring into existence”:

This word consists of five signs: three sound-signs and two determinatives. The first three signs are the coiled rope, pronounced w, followed by two loaf-of-bread signs, pronounced t. Thus, the word is transliterated as wtt and pronounced something like “wetet.”

The next two signs are determinatives and give the sense of the word. The first determinative is a hieroglyph that’s easily recognizable in any era.

It’s a penis, in case you didn’t notice. The penis glyph’s function in indicating the semantic meaning of “beget” is obvious. This sign is actually used in many Egyptian words, such as:

Yes, the penis hieroglyph can mean “thick” in Ancient Egyptian. I guess the priests who came up with this writing system wanted everyone to know a little something about their assets.

Now, back to “beget.” The second determinative in “beget” is the rolled scroll.

The scroll is often used for abstract concepts. This is because abstract concepts are often not easily represented by pictograms, but can be written down on, for example, a scroll.

Putting it all together, the combination of glyphs rope, bread, bread, penis, scroll produces a verb pronounced something like “wetet,” and meaning “to beget.”

Convergent evolution: Hieroglyphs are still used today

Here’s another fun fact about hieroglyphs. By pure chance, many modern emojis look nearly identical to their ancient counterparts. This has some wonderful examples of convergent glyph evolution, reproduced here for convenience.

And so, the next time one of your friends sends you a message like this:

I hope that you can appreciate it (syntactically, if nothing else) as a modern reinvention of an ancient form of writing.

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Upon seeing this message, most English speakers will wonder what the hell Sunil and I are talking about.

It has to do with a book he’s reading, called Alex’s Adventures in Numberland: Dispatches from the Wonderful World of Mathematics. Chapter one covers counting systems used in various societies — the Arara in the Amazon count in pairs, the Revolutionary French tried to make clocks count by tens and the Babylonians counted in base 60. But the most interesting counting system, to me, was the one used by shepherds in Lincolnshire, England, to count sheep.

1. Yan
2. Tan
3. Tethera
4. Pethera
5. Pimp
6. Sethera
7. Lethera
8. Hovera
9. Covera
10. Dik
11. Yan-a-dik
12. Tan-a-dik
13. Tethera-dik
14. Pethera-dik
15. Bumfit
16. Yan-a-bumfit
17. Tan-a-bumfit
18. Tethera-bumfit
19. Pethera-bumfit
20. Figgit

So when Sunil told me that covera pimp dik bumfit and bumfit pimp dik was 69, all he really said was that 9 + 5 + 10 + 15 + 15 + 5 + 10 = 69, which is true.

I find this counting system fascinating, and not just because counting pimp, dik, bumfit, figgit is hilarious and fun.

First of all, you’ll notice that this system is a hybrid base-five, base-twenty counting system. You have unique words up to ten, then compound words (Tan-a-dik = Tan + dik = 12) up to 15 (bumfit), then some more compounds with bumfit up to figgit (20).

Secondly, this counting system felt weirdly familiar to me. Yan and one, tan and two, tethera and three, pethera and four. What about dik? Well, this is clearly similar to dec, the Latin root for ten (French is dix, Spanish is diez, Italian is dieci). Even figgit looked familiar — the Latin ��ī�����Գ�ī, meaning 20, sounds a lot like figgit. My first thought was that this system is some kind of corrupted Latin, mixed with whatever Celtic language existed in Lincolnshire before the Roman conquest.

I wasn’t right about this, but I was close.

Consonant shifts and Proto-Indo-European

Why does pethera, which begins with a “p,” sound familiar to four, anyway?

Consonant shift! Linguists have discovered regular patterns of consonant shift that occur as languages evolve. The most famous of these sound shifts are the shifts that transform into its daughter languages (Latin, English, Sanskrit, Persian, etc.).

states that the Proto-Indo-European consonants underwent predictable, regular evolution as they evolved into Proto-Germanic and Germanic daughter languages.

For example, the Proto-Indo-European word for “brother,” ��ʰ��é��₂tŧ�� (something like “breh-ter”) evolved into the Proto-Germanic ����ōþŧ�� (“b����-�ٳ���”), and eventually into the Old English ������þ�ǰ� (“b����-�ٳ�ǰ�”).

By the way, that funny letter þ is called , which is an Old English letter pronounced “th.” If you had to read in high school English class, you might remember seeing þ all over the place.

“Father” is another good example of regular consonant shifts. Proto-Indo-European *���₂tḗr (“peh-ter”) evolved into Proto-Germanic *�ڲ���ŧ��, and eventually Old English ��æ����.

So “p” and “f” are linguistically very similar, especially in a Germanic language like English. Pethera and four could easily be derived from a common Indo-European ancestor.

The idea is similar to ��ī�����Գ�ī (Latin) and figgit (Lincolnshire shepherd’s dialect). The “f” and the “v” are very similar sounds, followed by the “g” and “t” sounds. Try pronouncing “vigint” ten times fast and see if it morphs a little into “figgit.”

It was at this point, while googling consonantal shifts, that I found this video from Numberphile, with one of the least searchable titles I’ve ever seen. From Numberphile, I present the gloriously titled :

In the video, Professor Roger Bowley says that the yan-tan-tethera number system is Celtic and predates the Roman conquest of Britain. So my theory of corrupted Latin is wrong — actually, both Latin and this obscure Celtic dialect have a common ancestor in Proto-Indo-European!

This explanation of the yan-tan-tethera origin fits much better than mine does. Wikipedia has a whole list of different variations on the yan-tan-tethera for various English regions.

Apparently, this weird-ass counting system is actually a very old counting system that probably predates the Roman conquest of Britain, and it’s linguistically related to all the other Indo-European languages! Some of the words are even the same!

But wait, what about bumfit?

Consider the bumfit, and make sure it’s hovera covered

Bumfit is a hilarious word. However, I don’t think “bumfit” sounds like “fifteen” at all. Nor does “hovera, covera” sound like “eight, nine” in any way. But if all the numbers in the yan-tan-tethera counting system are derived from Proto-Indo-European, how did eight and nine (*��₽�ḱt���₃ and *��₁né�ܲ� in Proto-Indo-European) become hovera, covera?

The explanation from the same says that bumfit and the rest are Proto-Celtic numerals that died out in modern English. The Welsh numerals do have something in common with the yan-tan-tethera system:

The Welsh pymtheg is … sorta similar to bumfit, I guess? And the Welsh pump, deg, pymtheg, ugain is at least partially recognizable as pimp, dik, bumfit, figgit.

The Ancient British word for twenty, �ɾ��첹�Գ�ī, is essentially identical to the Latin ��ī�����Գ�ī (remember, in classical Latin, “v” is pronounced “w”), so I guess the Wikipedia page’s claim that multiples of five are highly conserved checks out.

But this hypothesis seems somewhat lacking to me. Where do you get hovera (8) and covera (9) from? The Welsh versions are wyth and naw, and the Ancient British versions are oxtu and nawan. That’s not even close.

Counting Rhymes

Another friend of mine, Jill, mentioned to me that she had just finished reading The Writing of the Gods: The Race to Decode the Rosetta Stone, and the book had mentioned that the children’s nursery “Hickory Dickory Dock, the mouse ran up the clock” was originally a .

Short, common words, learned early in life, tend to be the most constant throughout language evolution (“mama,” “father,” “brother,” etc.). In the same way, counting rhymes, taught to children at a young age, are highly conserved linguistically.

This led me to the fantastic “The Secret History of ‘Eeny Meeny Miny Mo,’” by Adrienne Raphel, on the origin and history of counting rhymes. Seriously, give this article a read; it’s fascinating.

I would venture a guess that pretty much every English-speaking schoolchild knows some version of the rhyme:

Eeny, meeny, miny, mo
Catch a tiger by the toe
If he hollers, let him go
Eeny meeny miny mo

This rhyme has a darker history than I knew. According to Adrienne Raphel:

In the canonical Eeny Meeny, “tiger” is standard in the second line, but this is a relatively recent revision. If it doesn’t seem to make sense, even in the gibberish Eeny Meeny world, that you’d grab a carnivorous cat’s toe and expect the tiger to do the hollering, remember that in both England and America, children until recently said “Catch a nigger by the toe.”

Didn’t know that one. Yikes. But it seems that this is a fairly recent revision of a much more ubiquitous class of counting rhymes. In Denmark:

Ene, mene, ming, mang,
Kling klang,
Osse bosse bakke disse,
Eje, veje, vaek.

And in Zimbabwe:

Eena, meena, ming, mong,
Ting, tay, tong,
Ooza, vooza, voka, tooza,
Vis, vos, vay.

However, while reading this article, one particular rhyme caught my eye.

In 1830, children in Scotland chanted:

Zinti, tinti,
Tethera, methera,
Bumfa, litera,
Hover, dover,
Dicket, dicket,
As I sat on my sooty kin
I saw the king of Irel pirel
Playing upon Jerusalem pipes.

In that rhyme, found in Scotland, we see “tethera, methera, bumfa, hover, dover, dicket,” all recognizable yan-tan-tethera numbers. Raphel goes on to connect this counting rhyme to the same yan-tan-tethera counting system we’ve been discussing, which she gives as:

Yan, tan, tethera, methera, pimp,
Sethera, lethera, hothera, dovera, dick,
Yan-dick, tan-dick, tether-dick, mether-dick, bumfit,
Yan-a-bumfit, tan-a-bumfit, tethera bumfit, pethera bumfit, gigert.

Now I see what’s going on. The yan-tan-tethera counting system is much more than simply a linguistic evolution of the ancient Proto-Indo-European numbers; it’s a counting rhyme! Likely, it is designed to be a memory aid for a nonliterate population that needs to count things.

Some of the numbers are the same as ours — multiples of five, especially, are conserved from their Proto-Indo-European roots, but the system as a whole is meant to roll off the tongue as a rhyme, as unforgettable as “eeny meeny miny mo.” In fact, the children’s nursery rhyme “Hickory Dickory Dock” probably has its in this ancient Celtic counting rhyme, via the numbers “hothera dovera dick.”

The reason the yan-tan-tethera numbers are so fun to say out loud is the same reason that epic poetry is written in rhyming meter — repetitive, rhyming lines are very easy to memorize, which is enormously important for primarily oral cultures.

This really blew my mind.

It turns out that the yan-tan-tethera counting system really was familiar to me, and probably you too — every schoolkid in America already knows it as “Hickory Dickory Dock,” though its origins as a Proto-Celtic counting system are long forgotten.

[Dylan Black first published this piece on .]

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The views expressed in this article are the author’s own and do not necessarily reflect 51�Թ�’s editorial policy.

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And I found that a serpent was coming towards me. It was thirty cubits in length, and its beard was more than two cubits in length, and its body was covered with gold scales, and its eyebrows were of pure lapis lazuli…

And it opened its mouth to me, as I was lying flat on my stomach before it, and it said unto me, “Who hath brought thee hither? Who hath brought thee hither, O miserable one?”

That is from the Tale of the Shipwrecked Sailor, the oldest complete story that has come down to us from antiquity. It was written in Hieratic during the Middle Kingdom of Ancient Egypt, and I read it in the original Egyptian from the comfort of my bed, four thousand years after it was written.

Reading an ancient story is an experience unlike any other. It is a glimpse into the mind of an alien — utterly foreign, yet oddly familiar. A yawning gap of time, culture and language divides me from the author, but I read his tale nevertheless, and I marveled when the sailor encountered the serpent god with eyebrows of lapis lazuli.

I’ve often felt the pressure to better myself, and I occasionally crack a technical manual or a literary classic in my spare time in deference to that pressure. I’ve known colleagues whose hobbies are essentially identical to their work — Nothing would depress me more, though I’d probably be better at my job.

Instead, my hobbies are almost militantly useless. I write a blog in which I the hot-dog-ness of various sandwiches, and spin density waves if they were made of guinea pigs. The only foreign languages I speak are Latin and Ancient Egyptian, into which The House of the Rising Sun (so useful). The history books I love best are the furthest removed from my own time. In general, my delight in a hobby is inversely proportional to its utility.

But despite, or perhaps because of, the unavoidable pressure to be productive that pervades modern life, I feel that my useless hobbies are not only personally valuable, but essential to a life well-lived, and I think the ancient philosophers tend to agree with me.

Crawling in the mud: Zhuangzi and ��ú��é��

��ú��é��, nonaction in accord with the natural flow of the universe, was praiseworthy to the Daoist sage Zhuangzi, born 2300 years ago in ancient China. Zhuangzi was renowned across China for his wisdom, and his counsel was greatly desired by the political elites of the time. When the duke of Qi, one of many desiring wise counsellors, invited Zhuangzi to become his chief minister, his messengers found the old sage fishing among the river reeds. Upon receiving this job offer, Zhuangzi did not look up from his rod and :

“I have heard that there is a sacred turtle in Chu that died three thousand years ago. The duke keeps it in a casket wrapped in cloth and has placed it in a temple. May I inquire whether the sacred turtle wanted to be dead and to have its bones venerated by man? Or was its intention to stay alive and crawl around in the mud, dragging its tail?”

“Naturally,” replied the messengers, “it hoped to crawl around in the mud, dragging its tail.”

“Go home,” said Zhuangzi, “I also want to crawl around in the mud, dragging my tail.”

The serenity of Epicurus

, too, understood the value of nonproductive pursuits. He was a Hellenistic philosopher who suffered from chronic pain all his life, and perhaps fittingly, developed a philosophy focused on pleasure and pain. To Epicurus, what is true pleasure? True pleasure is not the fleeting pleasures of wealth, rich food and debauchery, but a restrained, mental satisfaction that lingers, like heat from the embers of a hearth. True pleasure is the absence of pain and freedom from unnecessary desires.

This pleasure comes with freedom from the desire for wealth, freedom from the fear of death and of the gods, the bond of the tight-knit community and from pure intellectual exploration — this pleasure is serenity, ataraxia in Greek. The fear of death was simply one more pain to overcome in the life of Epicurus, and so his followers wrote thusly on their tombstones: Non fui, fui, non sum, non curo — I was not, then I was, I am no more, I do not mind. 

Epicurus reminds us that the pursuit of wealth does only so much to decrease the pain of life. For indeed, what shall it profit a man, to gain the world but lose his soul?

Aretḗ and the joy of useless excellence

But for me, there is still more to life than ��ú��é�� and ataraxia, for there is pleasure too in purpose. Former US President John F. Kennedy, another great philosopher, spoke to this purpose when he of the Apollo moon mission that we choose to go to the moon not because it is easy, but because it is hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one we are willing to accept, one we are unwilling to postpone and one we intend to win.

Indeed, to strive for one’s excellence, for one’s �������ḗ, in any field of human endeavor, for no practical benefit whatsoever — this is the highest pursuit of man, his virtue par excellence, and his greatest good. To strive for mastery in a field, regardless of practical benefit and indeed in spite of it, is a noble and fulfilling pursuit.

So why do I waste my time? To exist in non-action, to be free of the burden of utility, to find ataraxia, to strive for something difficult, yet not lose myself in pursuit of material gain, and because ultimately, it is my time to waste.

In the words of Seneca, omnia aliena sunt, tempus tantum nostrum est — All else is foreign to us, only time is ours. And I intend to spend mine generously, spiced with those useless hobbies that bring me joy.

[Dylan Black first published this piece on .]

[ edited this piece.]

The views expressed in this article are the author’s own and do not necessarily reflect 51�Թ�’s editorial policy.

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The ancient Greeks were, quite reasonably, concerned by this question, because their democracies died all the time. In fact, this happened so much that the most eminent philosophers and historians of the classical period developed a theory that rationalizes the rise and fall of democracies, oligarchies and tyrannies. In this article, we will investigate whether their theory, called anacyclosis, holds up under scrutiny, and by scrutiny, I mean Monte Carlo simulations of government transition based on historical data from 1,035 Greek city-states. But first, some history.

The poleis of Ancient Greece

The peculiarities of classical Greece make empirical theories of political revolution much easier to imagine than in, say, the Persian Empire, which was a hereditary monarchy for pretty much its entire history. The dominant mode of social organization in the archaic and classical Greek periods is the polis, the city-state. Usually, there’s an independent mother city (Athens, Sparta, etc.) that politically, economically and culturally dominates its surrounding hinterland. Each city has its own constitution, or form of government, but shares a common Greek culture and language with its neighboring poleis.

This social structure is as dynamic as it is unstable, and there were many political revolutions. The ancient world’s most sophisticated theories of political evolution grew out of this dynamic — they classify government into a few categories based on which group holds power, and posit that they devolve sequentially from higher to lower forms. Let’s take a quick look.

Governmental types in Ancient Greek thought

The Inventory of Archaic and Classical Greek Poleis, about which we will have much more to say later, gives a nice introduction to the types of constitution.

In Greek political theory politeiai [political institutions] were divided into types according to how many people constituted and manned the principal organs of government. Basically, there were three constitutional types: the rule of the one, the few and the many. Pindar is the first we know who distinguished between rule by a tyrant, or the wise, or the whole army. About a generation later, Herodotos has a debate about the three basic types of constitution, here described as demos, oligarchia and monarchia. [In the early 4th century BC], Plato called the three forms tyrannis, aristokratia and demokratia.

Linear evolution in Plato’s Republic

Plato made a finer distinction, dividing government into five categories of constitution in his , and additionally giving their sequence of devolution.

Aristocracy (rule by the best) → Timocracy (rule by honor/worth/money) → Oligarchy (rule by the few) → Democracy (rule by the people/mob) → Tyranny (rule by one man).

Plato writes that governments devolve in this order, from best to worst, in a linear fashion, terminating in tyranny. I went back and checked the Republic to see if Plato makes any claims of a cyclical nature, and I don’t think that it does, but the Republic is very hard to read generally, so I’m not 100% sure.

Aristotle, a student of Plato, the tutor of Alexander the Great and a giant of philosophy in his own right, generally agreed with Plato, but distinguished between a good and a bad form of monarchy (basileia versus tyrannis), minority rule (aristokratia versus oligarchia) and majority rule (politeia versus demokratia). His conception, however, was also linear (as best as I understand).

Anacyclosis in Polybius’s Histories

Polybius was a Greek hostage and historian in Rome during Rome’s rise to power, and he improved upon Plato and Aristotle’s framework. Polybius divided government into three categories, each with a virtuous and corrupt form, for a total of six constitutions. They are as follows, from his :

The virtuous aristocracy is corrupted into an oligarchy, which is overthrown by the people as a democracy, which degenerates into mob rule or ochlocracy. A great leader emerges from mob chaos to create a monarchy, which descends into tyranny before being overthrown by the noble aristocracy, beginning the cycle anew. He called this cycle anacyclosis.

(to whom I must give credit for the genesis of this article), offers the following comments on this process:

There is good reason to think that Polybius and his predecessors arrived at this theory empirically. After observing the rise and fall of many hundreds of city-states, most of which cycled through several of the governmental forms mentioned above, Greek political thinkers concluded that these transitions from one form to another were not random. Rather, they seemed to follow simple and recognizable patterns. For example, tyrants were frequently overthrown by groups of aristocrats, while popular revolutions frequently overthrew oligarchies and ushered in democratic rule. Interestingly, the reverse of these trends (aristocracies being overthrown by tyrants or democracies turning into oligarchies) were statistically less likely to occur.

Through such observations, Polybius extrapolated the likely complete course of political evolution for an independent state whose lifecycle is not cut short by war or disaster.

Polybius, Plato and Aristotle essentially agree on the pattern — we go from rule-by-few (aristocracy/oligarchy) to rule-by-many (democracy) to rule-by-one (monarchy/tyranny), with an optional cycle back to rule-by-few.

Polybius thinks there’s a way out of this cycle. If one combines all three forms of government into a mixed constitution, a blend of democracy, aristocracy and monarchy, one can create a stable system exempt from anacyclosis. Polybius thought that the Roman Republic was the embodiment of this mixed constitution and the reason for its strength and longevity. The Founding Fathers of the United States of America, and John Adams in particular, were obsessed with Polybius and designed the structure of the United States government to avoid anacyclosis.

So to recap, we have several explicit claims, of which various authors claim subsets:

1. Political evolution follows a predictable pattern of oligarchy → democracy → monarchy.
2. This pattern may be linear (Plato) or cyclical (Polybius).
3. The reverse transitions are unlikely/unnatural.

Unfortunately for Polybius, he lacked the tools to quantitatively investigate his theory. Fortunately for us, we are much better than Polybius at linear algebra.

Political evolution is a Markov process

Implicit in anacyclosis is actually a fourth claim, the most important claim, that anacyclosis is “memoryless.” In other words, the next type of government depends only on the current type of government: Democracies always devolve to tyrannies, independent of what preceded democracy. In the theory of stochastic processes, this property is called the . We can use the Markov property to evaluate the validity of Polybius’ claim.

First, though, we need data. Fortunately, the Copenhagen Polis Centre has done most of the work for us, and An Inventory of Archaic and Classical Poleis, a monumental work that compiles the existing data/metadata on the 1,035 identifiable Greek city-states of the Archaic and Classical periods (c.650-325 BC). Among the data found in the Inventory is a list of city-states and their known government types, ordered by date.

This data is actually all we need in order to pretty fairly evaluate the validity of the theory! For each city-state, we can simply extract ordered pairs of government types from this list and count the frequency with which these transitions occur. Because anacyclosis is a Markov chain (remember that means memoryless), these transition frequencies completely define the system! Note that this method completely ignores staying in the same state as a transition of interest (which requires much more sophisticated data parsing). So this method will probe only when governments change between distinct types.

This will make more sense as we actually construct the transition matrix and learn how to analyze Markov processes more generally.

An introduction to Markov processes through the inventory of Greek poleis

A Markov process, named after the Russian mathematician , is a type of random process. It has discrete states and a notion of time. At each time step in the process, each state X has a probability ��(��→Y) of transitioning to state Y. The Markov (memoryless) property ensures that this probability is the only relevant characteristic of the system.

Markov processes are often represented by graphs that abstract these transition probabilities, like the one below from Wikipedia showing a two-state Markov process.

In essence, a Markov process is the simplest form of a probabilistic state machine that still has interesting behavior. For any process that can be assumed to be stochastic, or perhaps a system complex enough that its behavior approximates a stochastic process, we can model it as a Markov process and immediately extract nontrivial, useful properties (as we shall see later).

Processes that (approximately) have the Markov property show up everywhere. Weather prediction, stock price prediction and population genetics are all examples of approximately-Markov processes. In each of these cases, while the real system may have complex dependencies, a Markov model captures enough of the important behavior to be very useful while still being mathematically tractable.

So how would we construct a Markov model for political evolution in Greek poleis? Like any good scientists, the first thing we have to do is create a good visualization of our data, and stare at it.

The data

Looking at our dataset, we have six distinct types of constitutions listed by the Inventory. I’ll quote briefly from it here:

In the Inventory, when we classify the constitution of a polis, we distinguish between basileia, tyrannis, oligarchia and demokratia, but we ignore variants of the latter two types, and all attestations of basileia belong in the Archaic period…

In a few cases of serious doubt, we have used Mix. to describe a polis with an unidentifiable mixture of characteristics.

The inventory also has another category in the data not listed above, politeia, which Aristotle defined as the “good” form of democracy, but is also the general term for a “polity” in Greek. Both of these last forms, politeia and “mixed,” are very rare in the Inventory and slightly confusing.

We should also note that the Inventory says that the term basileia might change meaning over time, as it is only attested in the Archaic period and not the Classical period. Second, that “in actual fact, all polis constitutions were mixed,” to one degree or another.

But for a first cut, let’s ignore these complexities and take a look at the data. First, the total frequency of government types:

The total counts of each type of government in the inventory. I manually removed “klerouchy” and “dynasteia”, additionally (Pol.) in the very first line, because I’m unclear what it means. Each of these occurred only once.

We note that the mixed and politeia types are very rare and not likely to affect our results. Excellent! Let’s ignore them. Second, if we combine basileia and tyrannis, the constitutions are roughly equal in frequency between the autocratic, oligarchic and democratic categories. Interesting! We’ll keep the two types of monarchy separate for now.

Next, we can plot the frequency that any constitutional type appears first or last. If Plato is correct, we would expect to see lots of oligarchies initially and lots of tyrannies finally. We … might see some evidence for that? We really see more of a transfer between oligarchy and democracy from this graph, and the number of monarchies slightly decreases, but not a lot. I don’t think Plato gets much help from the data here.

Finally, we can plot the frequency of government transition types. We define this naively, taking the sequence of governments in the inventory, and plotting the frequency of each ordered pair, i.e., demokratia, oligarchia, basileia would count as one occurrence each of dem.→ol., ol.→bas..

This last plot is pretty much the key to Markov processes. We can simply reinterpret each column of the above heatmap as a probability of transition between states. Thus, by normalizing each column of the transition frequency heatmap, we get a transition matrix T that defines the process. Because our system has the Markov (memoryless) property, the single-step transition matrix entirely defines the process — it is a Markov process.

How does the model work in practice? Let’s enumerate the types of government as the ordered list:

[bas., tyr., ol., dem., mix., pol.]

Then we can define a one-hot vector v that corresponds to my state, e.g., [1, 0, 0, 0, 0, 0] = the system is in the basileia state. The probability of transitioning to any other state is then given by a vector p, equal to the transition matrix T times v.

Let’s explicitly construct T for our data. Since “mixed” and politeia are ill-defined and occur so infrequently, I feel fairly justified in simply dropping those columns from the data.

From this matrix, we can then recreate the Markov-process-style node graph.

Assessing validity

We’re now in a position to partially assess the validity of anacyclosis as it relates to the data, in a first-order sort of fashion.

The first claim is that political evolution proceeds in the order oligarchy → democracy → monarchy. Our data is fine-grained enough that we can split monarchy into its “virtuous” and “corrupt” forms, basileia and tyrannis, and so let’s look for oligarchy → democracy → basileia → tyrannis in the data.

Looking first at the basileia → tyrannis transition, we actually find excellent support for this in the data! Basileia to tyrannis transitions happen about 13 times more often than the reverse. However, there’s a confounding variable. Remember that the Inventory says that basileia is attested only in the Archaic period, so any basileia → tyrannis transitions might be the result of redefinition as opposed to transition. Let’s call this one a partial thumbs up, though.

How about the posited oligarchy → democracy transition? Not so much. These two nodes have the tightest connection in the graph, and the transition rates are essentially even, with oligarchy to democracy being ever so slightly more favorable than the reverse.

Finally, what about democracy → monarchy? It seems as though democracies don’t ever go to basileiai, which certainly doesn’t support the theory of anacyclosis, but again, this could be a definition thing — if we started the chain at basileia, we could have passed through the Archaic period before we got back around, when basileiai had turned into tyrannides. Unfortunately, though, the democracy → tyranny transition (29%) is much less common than tyranny → democracy (56%). This gets even worse if we consider basileia the same thing as tyrannis, which has even more asymmetry between the two transition frequencies. So the democracy to monarchy transition doesn’t find much support here; in fact, more the reverse.

In fact, what is the most common cycle? Let’s re-plot the Markov chain where we combine tyrannis and basileia into “monarchy.”

This doesn’t really help us much. The most plausible cycle by far is simply oscillation between democracy and oligarchy, which does not at all fit into the anacyclosis paradigm. It seems we require a more sophisticated analysis to extract the probable dynamics.

Markov chain Monte Carlo

Okay, well then, what does happen to a hypothetical average Greek polis? We can use a Monte-Carlo style simulation to find out.

“Monte Carlo” is a cute name for a very simple technique — if you have the rules of a system and want to understand its behavior, just simulate a whole bunch of random instances of that system and average the results. The simulation method is called “Monte Carlo” because one of the inventors an uncle who gambled too much in real Monte Carlo.

Nevertheless, this simple technique is extremely powerful. To implement, we

1. Choose an initial one-hot state v.
2. Multiply by our transition matrix to get p = Tv, our vector of state probabilities.
3. Choose a random state from p, weighted by the probabilities of each state, i.e., if p = [0.1, 0.2, 0.3, 0.4], we have a 10%, 20%, 30% and 40% probability, respectively, of choosing states 1, 2, 3, 4.
4. Repeat this for n timesteps.
5. Repeat steps 1-4 for m simulations.

Let’s try this out, keeping the basileia/tyrannis distinction, just for fun.

So keeping basileia as a separate category doesn’t do much, its occupancy fraction immediately hits zero (on average) and never recovers. The most distinct feature by far appears to be the oligarchy-democracy oscillation, settling after ten timesteps into an even mixture of democracy and oligarchy (remember this is an average; at each timestep, the system can only be in one state). The tyrannis initialization appears to cause the settling to happen faster, but doesn’t differ in the essential trend. We also appear to stabilize at a steady state for any initialization parameter! Perhaps this is the fabled “mixed” constitution that Polybius thought made the Roman state so powerful and stable? We shall formalize this thought later.

Before we do steady-state analysis, we should check for common cycles. Let’s plot the most common cycles we find, dropping the basileia/tyrannis distinction (both are monarchy) for clarity. 1000 more simulations…

The most common cycle by far is the 2-state democracy → oligarchy → democracy cycle. The next most common state is this same cycle, twice in a row!

But wait, if we look down the list at the fourth most common cycle, it’s democracy → monarchy → oligarchy → democracy, that’s anacyclosis! It is the most common three-state cycle! We found it!

Did we just prove anacyclosis is real? Well, uh, it depends on what you mean, I guess.

Instead of answering the above, difficult question, I choose to reinterpret the original theory in light of the data — when Polybius wrote that “anacyclosis consists of predictable, cyclic transitions from democracy → monarchy→ oligarchy,” he clearly must have meant that given the empirical transition probabilities derived from a Markov model of Greek city state constitutional data, a Monte Carlo simulation will show the most common three-state cycle is democracy → monarchy → oligarchy.

I think this is a very reasonable translation of the original Ancient Greek.

Mixed states and the stable distribution

Now, Polybius was particularly interested in ways out of this endless cycle. How can we find a stable governmental state? Both our answer and Polybius’ answer are the same, and are already hinted at by the results of the simulations. We noted earlier that the simulations, regardless of initialization state, seemed to settle into a predictable distribution of government types, roughly 40% oligarchy, 40% democracy and 20% tyranny. This was also Polybius’ answer, that a mixed state was a stable point of this Markov process.

Polybius took as his example par excellence the constitution of the Roman Republic, which had popular assemblies (democracy), the Senate (oligarchy), and two consuls with executive power (a dash of monarchy). He felt that this mixture was much more stable than any pure state, and lent Rome its fabulous power.

According to our model, Polybius is absolutely correct.

How can we find the stable state of our Markov process? Well, first, we extend the model to allow for mixed states by not forcing our state v to be one-hot. That’s fairly easy. But how do we find the long-term stable state, if there is one?

Let’s think geometrically about our transition matrix. For a 2 × 2 transition matrix, we can visualize its action by seeing how it transforms a set of vectors arranged along the unit circle in the plane.

Under this linear transformation, the unit circle becomes an ellipse. The special directions that remain unchanged (up to scaling) by this transformation are called eigenvectors. These are precisely the principal axes of the resulting ellipse.

Mathematically, an eigenvector v with eigenvalue λ is defined by the equation:

Where T is our transition matrix, but this is very helpful for us, because if I now apply T twice, I get:

Thus, when we apply T repeatedly, each application multiplies the magnitude of v by λ while preserving its direction.

This means:

• If λ = 1, v maintains its magnitude: it’s a stable state.
• If |λ| > 1, v grows without bound.
• If |λ| < 1, v shrinks toward zero.

For Markov transition matrices, the guarantees that 1 is always an eigenvalue, and all other eigenvalues have absolute value strictly less than 1, meaning they decay to zero after a long time. When the Markov chain is also , meaning that you can visit any state from any other state, and you never get stuck in deterministic cycles, this unit-eigenvalue direction corresponds to a unique stable distribution called the .

Any initial distribution will converge to this stationary distribution as we repeatedly apply the transition matrix. We actually saw this in our Monte Carlo simulations — did you notice how, no matter the initial state, we always ended up with the same fraction of oligarchy/democracy/monarchy?

So we can quite quickly find the stationary distribution of our transition matrix by performing an eigendecomposition of our transition matrix. We solve:

For all v and λ.

Let’s take our transition matrix, where we combine basileia and tyrannis into monarchy.

The eigendecomposition of T yields three eigenvectors.

Which does indeed have a unit-valued eigenvector! To find the stationary distribution, we only need to normalize the eigenvector with eigenvalue 1, by dividing it by its column sum (it’s a probability vector, remember).

We can visualize this final distribution with a bar graph. Bar graphs are the most useless graph type, but they are visually arresting due to large bars of solid primary colors, so I’m making one.

This graph shows the anacyclotically stable distribution of the Markov poleis model. I don’t know if the word anacyclotically will ever really catch on in popular discourse, but I think it really rolls off the tongue. Perhaps instead we should call it the Polybian distribution. It does look remarkably similar to the Roman system, which was deeply suspicious of kingship but recognized its utility, and hence had two equal consuls in the place of one tyrant, as well as theoretically balanced popular assemblies and an aristocratic/oligarchic Senate.

So that’s it! This is the final confirmation that Polybius was on the right track, and that if he had only been better at linear algebra, he could have quantitatively estimated the proportion of democracy, monarchy, and oligarchy to inject into a politeia to stabilize it against the inevitable anacyclosis, assuming of course that by anacyclosis he actually meant the stochastic Markov process we’ve been working with this whole time, and not the actual anacyclosis that he wrote down, which is given minimal support by the actual data. Easy!

I should note, for future work, that there is at least one major oversight — for any given year in even the most fractious Greek polis, the probability of government transitioning to an entirely new category is small. In other words, I could have structured this process around a timestep being a single year, instead of an arbitrary “government transition time,” and gotten very different-looking processes, with the same long-run transition probabilities. Oh well, you always have to leave work for the next researcher.

Addendum: methodological validity

Mere minutes after posting this article, I had a thought — is the Greek poleis data set even capable of detecting an anacyclosis cycle in principle?

Let’s say I have a sequence A → B → C → A → …, like our poleis dataset. Then, because of spotty recordkeeping, let’s say I decimate this sample by randomly deleting entries, so maybe I’d get A → B → __ → A → … Without knowledge of the original sample, there’s a spurious B → A transition in our data!

So the question I want to ask here is, given a sequence composed of a pure cycle S = A → B → C → A …, if I randomly sample this sequence by throwing away all but a fraction f of the data points, can I still detect my sequence above noise?

Formally, let’s ask the question in the following way: I sample a fraction f of the data points from my sequence S, and construct a Markov transition matrix T by naively measuring transition frequency between neighbors in my sampled sequence. How often will I measure that the probability of the original sequence is greater than that of the reverse sequence? This is a sensible definition of “noise” because if we are solely interested in three-element sequences with unique elements, there are exactly two, A→B→C and C→B→A.

This sounds like an interesting analytical problem, but keeping with the Monte Carlo theme of this article and my own laziness, let’s just try it in code. I’ll construct a sequence of length 100 (A→B→C→…), decimate it, keeping a fraction f of the data, construct my matrix T, and then check whether the original cycle probability exceeds the reverse cycle probability.

The results are as follows.

And for completeness, does this give sensible results when run it on the reverse sequence C → B → A? In other words, what’s the spurious detection rate for a sequence that doesn’t contain the cycle at all?

Yes. This test does indeed give sensible results.

How about a random sequence? What do we expect on average?

Yes, this also makes sense. For a random sequence, we detect that A→B→C is more probable than the reverse about 50% of the time. I think this is a , where I correctly detect that the probability of my sequence is higher than the reverse sequence (the noise), but this doesn’t mean anything, because the underlying generator is fully random.

Based on this, I think I can say that the cutoff point where this detection mechanism starts to work is when the probability of detection exceeds ~50% on that curve, so let’s say a sampling fraction of about f = 0.2. That’s not too bad. I think I can consider my method valid enough for a Substack article.

[Dylan Black first published this piece on .]

[ edited this piece.]

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