Ancient-Greek Numberal Systems
A few days ago I stumbled over this weird character ϡ I now know its called a sampie :) . So I hopped on wikipedia in order to read the first paragraph of its page, get some brief closure and then forget about this random factoid forever. But unbeknownst to me, I had just fallen down the medium-sized rabbit hole that are Greek numerals. Now I have a lot of “knowledge” in my brain about somthing called the Acrophonic, Thesis and Milesian system and I would love to offload it my brain is small and upgrading is just not viable with current memory prices on this page.
| Character | Acro- phonic |
The- sis |
Mile- sian |
|---|---|---|---|
| α - Alpha | – | 1 | 1 |
| β - Beta | – | 2 | 2 |
| γ - Gamma | – | 3 | 3 |
| δ - Delta | 10 | 4 | 4 |
| ε - Epsilon | – | 5 | 5 |
| ϛ - Stigma | – | – | 6 |
| ζ - Zetta | – | 6 | 7 |
| η - Etta | 100 | 7 | 8 |
| θ - Theta | – | 8 | 9 |
| ι - Iota | 1 | 9 | 10 |
| κ - Kappa | – | 10 | 20 |
| λ - Lambda | – | 11 | 30 |
| μ - Mi | 10000 | 12 | 40 |
| ν - Ni | – | 13 | 50 |
| ξ - Xi | – | 14 | 60 |
| ο - Omicron | – | 15 | 70 |
| π - Pi | 5 | 16 | 80 |
| ϟ - Qoppa | – | – | 90 |
| ρ - Rho | – | 17 | 100 |
| σ - Sigma | – | 18 | 200 |
| τ - Tau | – | 19 | 300 |
| υ - Ypsilon | – | 20 | 400 |
| φ - Phi | – | 21 | 500 |
| χ - Chi | 1000 | 22 | 600 |
| ψ - Psi | – | 23 | 700 |
| ω - Omega | – | 24 | 800 |
| ϡ - Sampi | – | – | 900 |
The Acrophonic Approach
The acrophonic system (also known as Attic numerals) got its name from the fact that the numerals are, well … acrophonic. This attribute is useally used for alphabets in which each letters name starts with said letter (for example the greek alphabet: α -> άλφα (alpha), β -> βήτα (beta), γ -> γάμμα (gamma), δ -> δέλτα (delta) … ). In our case it is a reference to the derivation of the numerals, where a numbers numeral is the first letter of the number (i.e. 10 -> δέκα/déka -> Δ). Somewhat similar to the roman system, there are numerals for the numbers 1(Ι), 5(Π or 𐅃), 10(Δ), 100(Η), 1000(Χ) and 10000(Μ), aswell as for the multiples of five 50(𐅄), 500(𐅅), 5000(𐅆) and 50000(𐅇). The later are just a fusion of the numeral 𐅃 and the powers of 10. Numbers are then represented via a composition of those base numerals (for example: 42 -> ΔΔΔΔΙΙ, 2025 -> ΧΧΔΔΠ).
To be completly honest, this system is not too shabby. It looks relativly nice, although it is plagued by the repitition of many numerals since there is no concept of digits. I would rate it a solid ΠΙ out of Δ.
The Thesis Approach
The thesis-principle is far easier to understand, even though it uses 140% more numerals than the arabic or acrophonic system. This is easily explained, since each of the 24 letters of the greek alphabet gets a value between 1 and 24 acording to its place (θέσις/thésis) within the alphabet. Since it only encompases numbers from 1-24, it was not a very usefull tool for mathematicians. Instead, these numerals where used as ordinals, for example to index the 24 rhapsodies of the Iliad/Odyssey, or as a way to mark the neighborhoods of Alexandria.
The lack of complexity is a two-edged blade. Whilst easy to learn, the thesis system has a relativly narrow usecase. Therefore on a scale of ω, I bestow it a score of λ.
The Milesian Approach
Compared two the previous two systems, understanding the milesian approach should be more intuitive, since, similar to the arabic/indian system we use today, it is based on a 9-digit approach. But there is a catch. Unlike the arabic system, the unit’s, ten’s and hundred’s do not share the same set of 9 numerals, because that would be boring. In the milesian system each power of 10 well, only the ones from 1 to 100 has a different set of unique digits. The alphabet is chopped into sections of length 9, which results in 3 sets of numerals that represent the numbers from 1-9, 10-90 and 100-900. To depict a number between 1-999, one only needs to chose a numeral from each set, so they sum up to the target number (20 -> κʹ, 777 -> ζοψʹ). In order to distinguish the numbers from regular words, they where overlined (φνα) in handwritten text, although with the invention of the printing press the overline was replaced with a trailing apostroph-like character (φναʹ).
But there was a problem: At the time, the greek alphabet had only 24 of the 3 * 9 = 27 letters required to make this system work. In order to solve this, three symbols from past/revised alphabets where taken in order to fill in the gaps. These symbols where:
- The Stigma ϛ / Digamma ϝ
Short history of the stigma
The digamma was the sixth letter of the greek alphabet before it was discontnioud around 500 BCE, before getting a revival as the numeral for the number 6. A few centuries down the line, the digamma had gone through a bit of a transformation->
->
->
->
->
and had actually become quite conflated with the ligature of the sigma-tau
σ +
τ ->
digraph (called stigma
closed and open variant of the stigma (ϛ) ). Therefore, the stigma replaced the digamma in the milesian number system. Funnily enough, the use of the stigma as representation for the στ and a numeral is still visible in modern Greece, where the ordinal representation for 6 is ΣΤ'. This closes a large circle for me personally, since as a kid, I never got why the ε-th grade was not followed up by the ζ-th grade.
- The Qoppa ϟ (alt. ϙ)
Did you know this about qoppa?
I could not find any interesting stuff about this character, but I wanted to add this section for consistency reasons.
- The Sampie ϡ (alt. Ϡ)
Want a sampie-themed knowledge nugget 🤨
Because large numbers where more uncommon, it is particularly hard for researches to pinpoint when sampie was introduced as the numeral for 900. Athenians where the last of the greek states to adopt the milesian system. The full adoption happened around 50 CE, but the first confirmed use of a sampi does not appear until 200 CE.
I like the trade-off of the milesian system in regards to efficiency vs. generalization: Some numbers requiere a lot less digits (|900| = 3 vs. |ϡ| = 1), but you need a new set of numerals for each additional power of 10. One way to get around this, is to use a modifier on the original 27 numerals and repurpose them. In order to expand the admittedly small number space from 1-999 to 1-999999, a leading lowered apostroph was introduced to show that a numeral had been mutlitplied by 1000 (͵ακδʹ -> 1024).
Click here for a fun "fact"
In medievel copies of the Book of Revelation, the number of the Beast was written like this χξϛ. I think the arabic numerals just hadn't caught on yet, but according to my head-canon writers wanted to avoid bad juju and thought Satan could not understand greek.
Unfortunatly, despite this nifty workaround, I think the downsides outweigh the benefits (there’s a reason why we have adapted the arabic/indian system). But even though it ends up losing in a direct comarison to the arabic/indian system, it is still a very effective numbering system with a neat philosophy behind it. I rate it a ϡιϛʹ out of ϡϟθʹ.
There is actually a lot more interesting stuff to tell about the milesian system. For example its use by the cult of Pythagoras in the practice of onomancy (not to be confused with the 9th school of magic, “oh-no!"-mancy, the only magic school where sneezing during the conjuring of a fireball is actually encouraged), a form of fortune telling where an individuals name was transformed into numbers, which where believed to hold divine meaning.
But I am tired, so this story will have wait for another day.
Sources: The wikipeia articles Greek Numerals (and its german counterpart Griechische Zahlzeichen), Milesisches System, Stigma aswell as two dozen more articles that are linked by the ones mentiones and the stigma images from wikimedia.