How to pronounce greek letters

Josef Stefan’s fourth-power law, written in both greek and english.
Author
Affiliation

Markus Agwin Kloimwieder

Scicloj

Published

November 27, 2025

The greek letters \(\xi\) (“xi”), \(\eta\) (“eta”) and \(\zeta\) (“zeta”) are often used in textbooks as counterparts to the latin x,y and z. This needs practice. All the more so that there also are the two greeks \(\chi\) (“chi”) and \(\omega\) (“omega”).

I’d like to make a table for this:

xi

eta

zeta

chi

omega

For practice, I start with a widely forgotten formula by Messieurs Dulong and Petit

\[\left({\eta}^{\xi} - 1\right)\,\zeta\]

How on earth do you pronounce that? Here we go

eta power xi minus 1 times zeta with xi in Celsius

and

eta zeta being constants

Included is the hint that the xi-\(\xi\) is a temperature measured in degree Celsius. But what are the constants eta-\(\eta\) and zeta-\(\zeta\)?

The formula of Dulong&Petit

I define a function which includes the sought-for numbers

(defn dp-formula [Celsius]
  (calcbox [((((eta power xi) minus 1) times zeta)
             with
             [xi in Celsius])
            [and [(1 comma 0 0 77) for eta]]
            [and [(2 comma 0 2) for zeta]]]))

This reads as

(kind/hiccup [:blockquote (:hiccup (dp-formula 0))])

eta power xi minus 1 times zeta with xi in Celsius

and

1 comma 0 0 77 for eta

and

2 comma 0 2 for zeta

The formula prints like this

(tex (:calc (dp-formula 'xi)))

\[\left({1.0077}^{\xi} - 1\right)\,2.02\]

I can now also calculate a number, e.g. for 100 °C

(:calc (dp-formula 100))
2.3298794924642854

But what does this number mean? It is degrees per minute. In 1817, Dulong and Petit measured the rate of change with time of the indicated temperature on a previously heated mercury-in-glass thermometer with a spherical bulb placed centrally in a spherical enclosure held at zero degrees Celsius.

These were their measured values

°C °C / min
80 1.74
100 2.3
120 3.02
140 3.88
160 4.89
180 6.1
200 7.4
220 8.81
240 10.69

So, when the thermometer was at 100°C, within the first minute it lost 2.3 degrees due to radiation. In that case, their formula was pretty accurate. But a man named Pouillet used the formula to estimate the temperature of the sun, got a value of some 1700 degrees, and that seemed pretty low to a certain Josef Stefan. He re-published the above data and proposed his famous fourth-power law on the relationship between heat-radiation and temperature on pages 391-428 of the “Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Classe, Neunundsiebzigster Band, Wien, 1879”.

The law of Stefan

(defn stefan-law [Celsius constants]
  (calcbox [((((chi plus xi) power 4 )
              minus
              (chi power 4))
             times
             omega)
            [with [xi in Celsius]]
            [and [[chi omega] being constants]]]))

\[\left({\left(\chi + \xi\right)}^{4} - {\chi}^{4}\right)\,\omega\]

chi plus xi power 4 minus chi power 4 times omega

with

xi in Celsius

and

chi omega being constants

We need to set omega-\(\omega\) to one six billionth. The other constant is given by the absolute zero temperature, chi-\(\chi\) = 273.

As an exposition, we calculate the fourth-power of 273. The result is a pretty big number.

chi times chi times chi times chi

with

chi equals 273 

5554571841

I can imagine that in the 19th century, without having computers, to fit some data it took considerable guts to take on a fourth power law.

(defn stefan-law-numbers [Celsius]
  (calcbox [((((chi plus xi) power 4 )
              minus
              (chi power 4))
             times
             omega)
            [with [xi in Celsius]]
            [and [(one (6 billion) th) for omega]]
            [and [273 for chi]]]))

chi plus xi power 4 minus chi power 4 times omega

with

xi in Celsius

and

one 6 billion th for omega

and

273 for chi

\[\left({\left(273 + \xi\right)}^{4} - 5554571841\right)\,\frac{1}{6000000000}\]

Stefan’s Law passes the first test in fitting the data as well as the old model.

°C °C / min D&P-formula Stefan law
80 1.74 1.71 1.66
100 2.3 2.33 2.3
120 3.02 3.05 3.05
140 3.88 3.89 3.92
160 4.89 4.87 4.93
180 6.1 6.01 6.09
200 7.4 7.35 7.42
220 8.81 8.9 8.92
240 10.69 10.71 10.62

With his new formula, Josef Stefan estimated the lower bound of the temperature of the sun to be around 5600 °C which means he was pretty much bang-on within some 100 degrees.

source: src/mentat_collective/emmy/josefstefan.clj