Zipf - Subtly Admiring Polymathy

Auerbach

Part 1 of our article series on Zipf[1] briefly touched on the pre–Zipfian history, including names such as Jean-Baptiste Estoup,[2] Godfrey Dewey,[3] E. U. Condon,[4] and Felix Auerbach.[5] This time, we're focusing on Auerbach (1856–1933). He was a professor of theoretical physics at the University of Jena, where he worked on things such as magnetism, hydrodynamics, and the hardness of solid materials. He also had a particular interest in mathematics, which may explain how, investigating the distribution of German city sizes, he arrived at something Zipfian three decades early.

Das Gesetz der Bevölkerungskonzentration

Indeed, he had already found a similar regularity for cities in "The Law of Population Concentration" (Das Gesetz der Bevölkerungskonzentration): rank a country's cities by population (largest = rank 1), then multiply each city's rank by its population, and what you'll see is that after the very largest cities, the product settles to a roughly constant value. Auerbach called that absolute Konzentration (A.K.; "absolute concentration" in Antonio Ciccone's 2023 translation[6]). For Germany in the 1910 census, A.K. from rank 15 on stays between 45 and 53 (in hundred-thousands). That's Zipf's law, written as rank × size ≈ constant instead of size ∝ 1/rank. To compare countries he divided A.K. by total population (expressed in hundreds of millions) and called the result spezifische Konzentration (Sp.K., "specific concentration"): a size-normalized measure of how clustered people are into cities. Concentration isn't the same as density, Auerbach argues: British India is almost twice as "dense" as Great Britain, yet Great Britain's concentration is eight times higher. Between 1895 and 1910 Germany's population density rose 23 percent, but specific concentration rose 40 percent, which means people were concentrating into cities faster than the country was growing. Off-topic, but whenever I see words like Bevölkerungskonzentration, I question whether German is a real language or I'm being ragebaited. The point is: Auerbach treated rank × size ≈ constant as the simple case of a broader rank-property regularity.

Mountains, Auerbach posits, when contrasting city scaling with other things, exhibit a slight rank-size distribution ("as the highest summit usually surpasses the following ones just by a little"), because they are formed by a single physical force that quickly exhausts itself. Personal wealth, on the other hand, demonstrates a stronger power-law relationship with α = 2, wherein a halving of the wealth threshold produces a fourfold increase in qualifying individuals, such that half-millionaires outnumber millionaires by a factor of four instead of a factor of two that a linear relationship would suggest (a Pareto distribution with α = 2, stated without citing Pareto[7]). Cities are somewhere in the middle of this spectrum.

Auerbach speculated that the more complex and multi-layered the forces driving a system, the stronger and more pronounced its power-law scaling will be. He was a physicist, not a geographer or economist, so his incursion into what we might call the beta version of econophysics is accordingly that of a natural scientist. Nevertheless, he explicitly eschews the deterministic application of inviolable natural laws to human affairs, while simultaneously arguing that human systems do follow laws, albeit those laws are predicated upon "more complex relationships" ("die größere Komplikation der Verhältnisse").

In effect, this is one of the founding documents of city-size power laws. Lotka[8] first cast the regularity in the modern rank-size form; Saibante[9] extended it across countries and time. Zipf[10] embedded it in his general power-law program, after which it became known as "Zipf's law for cities." But if Auerbach did it first, why did Zipf's name get promoted to canonical nomenclature while Auerbach's got relegated to the footnote section? The simplest explanation may simply be the language barrier, since Auerbach's original work was written in German, and as Ciccone points out in his translations' introduction, the paper, while recognized in modern literature, never received a publicly available translation prior to his efforts.

Omission

Rybski & Ciccone[11] reconstructed and measured this history. The quantitative bibliometric claim of their paper, Auerbach, Lotka, and Zipf: pioneers of power-law city-size distributions, is that the discovery gets almost entirely attributed to Zipf, which they realized by filtering Google Scholar to city-related citations only. They found 334 works citing Auerbach's 1913 paper versus 1,368 citing Zipf's 1949 work (227 citing both). Roughly 20% of city-related Zipf-citers also cite Auerbach; almost all Auerbach-citers cite Zipf. The authors offer three reasons beyond my (1) language-barrier proposition: (2) Auerbach used the regularity as a prerequisite for his concentration statistic, not as the finding; (3) he published it in a two-page paper in contrast to Zipf's 1949 monograph; and (4) early economics literature didn't cite it. They propose renaming "Zipf's law for cities" to the "Auerbach-Lotka-Zipf (ALZ) law." While it seems like the name hasn't caught on, the paper, alongside Ciccone's same-year translation, finally made a core text accessible in English after 110 years.

However, in a weird way, this neglect is reinforced by the fact that even Ciccone, the co-author of the history-of-science treatise and translator of the source paper, has numeric slips in his work that, as far as I know, have never been corrected. It's not that Ciccone was negligent, rather, the lack of peers to obsessively verify every printed number is alarming. Finding three versions of The Law of Population Concentration released by him (one from 2021 with a German transcription that repeats the same mistake,[12] one on his University of Mannheim page,[13] and the 2023 published version), all of which make the same error, suggests he worked from a mistyped or badly OCR'd transcription. Ciccone's German transcription prints "Die Zahl 47,2 ist also ... für die Bevölkerungskonzentration ... charakteristisch" with a population figure of "64,6 Mill.," whereas the 1913 scan shows 47,8 and 64,5, and arithmetic sides with the scan: 47.8 ÷ 0.645 ≈ 74.1, matching Auerbach's printed 74, while 47.2 ÷ 0.645 ≈ 73.2 would round to 73.

The Test, Briefly

Naturally, we had to figure out what these numbers actually mean, because my life is empty and I have nothing better to do. Here, I should mention that the recomputation wouldn't have been possible without the help of a few capable AI agents (namely Kimi K3, GPT-5.6 Sol, and Qwen3.8-Max). Their exact contributions are listed in CREDITS.md on the repository.[14] While the computation was done by AI, the verification was fully under human (debatable; it was me) control, so if there are any remaining errors, those errors are mine.

Auerbach's 1913 numbers hold up, but only with a textual correction. A double-entry transcription of his Table 1 (94 German places ranked by population) reproduces his "Konzentrationszahl" band of 45–53 from rank 15 onward, exactly as he reported, but the printed mean of 47,8 turns out to be an average across all 94 cities, not the stabilized tail mean his prose implies. The actual tail mean is near 50. A free-exponent fit on all 94 cities gives ξ = 0.9801, statistically compatible with the Zipf value of 1, though the 95% interval [0.7787, 1.1851] is too wide to separate nearby exponents.

On modern German data, the same broad rank–size shape stays, but concentration levels and country ordering have moved. Germany in 2025 still produces a Zipf-compatible exponent (ξ = 1.0798), while the A.K. band has shifted from 45–53 to 57–90. A.K. scales with city size relative to national population, and it seems a century of urbanization shifted it. The pre-registered test of Auerbach's twelve-country ordering finds a (fragile) concordance: Kendall τ = +0.5556 across the nine one-to-one successor states.

Comparing Functional Urban Areas to plain municipalities gives a specific-concentration ratio roughly 70% higher under the FUA definition, against the 4.05% Auerbach measured in 1910. The comparison is coarse and likely overstates a suburb-merging effect, but the direction is unambiguous.

For the mountain claim that was mentioned in Auerbach's article, that summit heights decline "much more gently" with rank than city populations do, the evidence splits by arm. The most scientifically accurate description I could provide would be "I guess, kinda true." In all ten tested arms, the rank curve declines more gently than inverse (ξ < 1). But pure power laws are rejected on absolute goodness-of-fit in every large arm at the selected cutoff. Bounded or truncated families win the model comparison on the global and Himalaya arms, while the Alps, Rockies, and highest-prominence tail satisfy the stricter rank-law lane. Summit-list coverage bias pushes every fitted exponent downward, toward Auerbach's direction, which means the drift across prominence thresholds is therefore evidence about the bias, but does not prove the claim.

Following our preregistered guidelines, Auerbach's results show a split verdict: his city band and stabilization rank hold up, while his specific-concentration level does not translate to contemporary data. His country ordering leaves a trace but no durable ranking. His mountain sentence needs some caveats. The complete analysis and end-to-end reproducible code pipeline are available in our GitHub repository[15] alongside the full report.[16]

Are we Auerback?

Auerbach's importance, since Rybski's short biography, Auerbach's Legacy,[17] is slowly being rediscovered. Rybski and Ciccone's ~20% figure is a generous rounding of 227/1,368 = 16.6%, with the intersection dominated by a recent wave of recognition. Fig. 3b shows values around 20% in 2000-2020 but lower for earlier decades; post-2020 literature does appear to credit him more frequently (Cottineau[18] finds most non-Zipf-crediting work predates 1990), and that bumps up the numbers somewhat. While this is not an academic journal, I hope to be a part of this wave at least in spirit, primarily because I find the story of an unrecognized polymath dabbling into a topic and casually proposing a law he himself doesn't even consider to be his main finding that later, formulated by someone else, gets referenced as "the most robust statistical regularity in all the social sciences," immensely fascinating.

Footnotes

[1] "Zipfvestigations Part 1," Cacozelia, August 30, 2026, https://cacozelia.com/posts/zipf-part-1/.

[2] J.-B. Estoup, Gammes sténographiques, 4th ed. (Paris: Institut Sténographique, 1916). The rank-frequency observations Zipf later cited appear in this edition's theoretical fascicule; earlier editions of the drill book (from 1907/1912) are often cited in error.

[3] Godfrey Dewey, Relativ Frequency of English Speech Sounds (Cambridge, MA: Harvard University Press, 1923).

[4] E. U. Condon, "Statistics of Vocabulary," Science 67, no. 1733 (March 16, 1928): 300, https://doi.org/10.1126/science.67.1733.300.

[5] Felix Auerbach, "Das Gesetz der Bevölkerungskonzentration," Petermanns Geographische Mitteilungen 59 (1913): 74–76.

[6] Antonio Ciccone, trans., "The Law of Population Concentration," Environment and Planning B: Urban Analytics and City Science 50, no. 2 (2023), https://doi.org/10.1177/23998083221147139.

[7] Vilfredo Pareto, Cours d'économie politique: professé à l'Université de Lausanne, 2 vols. (Lausanne: F. Rouge, 1896–97).

[8] Alfred J. Lotka, Elements of Physical Biology (Baltimore: Williams & Wilkins, 1925).

[9] Mario Saibante, "La concentrazione della popolazione," Metron: Rivista Internazionale di Statistica 7, no. 2 (1928): 53–99. (Rybski and Ciccone cite this as the piece that extends the rank-size rule across countries, regions, and time; an earlier Saibante article in Metron 6, no. 1[1926] on industrial capital concentration is a different work.)

[10] George Kingsley Zipf, Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology (Cambridge, MA: Addison-Wesley, 1949).

[11] Diego Rybski and Antonio Ciccone, "Auerbach, Lotka, and Zipf: Pioneers of Power-Law City-Size Distributions," Archive for History of Exact Sciences 77 (2023): 601–13, https://doi.org/10.1007/s00407-023-00314-0.

[12] Antonio Ciccone, trans., "Das Gesetz der Bevölkerungskonzentration — The Law of Population Concentration —" (working translation with introduction, March 2021), https://www.vwl.uni-mannheim.de/media/Lehrstuehle/vwl/Ciccone/auerbach_1913_translated_with_introduction_March_2021.pdf.

[13] Antonio Ciccone, trans., "The Law of Population Concentration" (University of Mannheim page copy), accessed September 5, 2026, https://www.vwl.uni-mannheim.de/media/Lehrstuehle/vwl/Ciccone/July_23_law_of_population_concentration_translated.pdf.

[14] "CREDITS.md," kenrinzero/auerbach-cities-and-mountains, GitHub, accessed September 5, 2026, https://github.com/kenrinzero/auerbach-cities-and-mountains/blob/main/CREDITS.md.

[15] "auerbach-cities-and-mountains," GitHub, accessed September 5, 2026, https://github.com/kenrinzero/auerbach-cities-and-mountains.

[16] "auerbach-cities-and-mountains," GitHub Pages, accessed September 5, 2026, https://kenrinzero.github.io/auerbach-cities-and-mountains/.

[17] Diego Rybski, "Auerbach's Legacy," Environment and Planning A 45, no. 6 (2013): 1266–68, https://doi.org/10.1068/a4678.

[18] Clémentine Cottineau, "What Do Analyses of City Size Distributions Have in Common?," Scientometrics 127, no. 3 (2022): 1439–63, https://doi.org/10.1007/s11192-021-04256-8.