Chapter 1The Shape of Empires
The charts only have data at dated waypoints; between them they draw a straight line, which means: assume the quantity changed at a constant rate. Rome’s extent has waypoints (60 BCE, 1.95 Mkm²) and (30 BCE, 2.75 Mkm²). What was Rome in 45 BCE — halfway through that interval?
The fraction (15/30) is just “how far through the interval are we?” — halfway, so take half of the rise. That is the whole of eq. (1.2). Go here for more.
Comparing economies at market exchange rates undercounts places where things are cheap. A haircut in Chicago costs $30; the same haircut in Chennai costs about $4 at market rates — yet it is the same haircut, the same “output.” PPP fixes this by repricing every country’s goods at one common set of prices before comparing.
PPP: both haircuts count the same → India’s real output share rises.
That is why China’s output share “re-passed” America’s around 2014 in PPP terms while remaining behind at market rates — both statements are true; they answer different questions (PPP: how much stuff; market: how much purchasing power abroad). These essays use PPP because empires’ power rested on the stuff. Go here for more.
To compare the shapes of two curves of very different sizes, divide each by its own maximum so both peak at exactly 100%. The British Empire peaked at 35.5 Mkm² in 1920; in 1945 it held 33.0 Mkm²:
After this rescaling Britain’s curve and the Mongols’ can sit on one chart and differ only in shape and tempo — which is what Figure 3 of chapter 1 compares. The cost: all absolute size information is deliberately thrown away. Go here for more.
Chapter 2A Common Currency for Power
The arithmetic mean adds and divides; the geometric mean multiplies and takes a root. The difference matters when the inputs are unbalanced. Take the United States in 2026 — shares of 4.1 (people), 14.7 (output), 37 (force), 33 (network):
geometric: (4.1 · 14.7 · 37 · 33)1/4 = (73 590)1/4 ≈ 16.5
The product drags the answer toward the weakest input: because each factor multiplies the rest, one small number scales the whole thing down, and no large number can buy it back. That is the “punishes lopsidedness” property the essay relies on — and 16.5% is exactly the U.S. value the interactive shows on equal weights. Go here for more.
In eq. (2.3) the weights sit in the exponents: a lens with weight 0.5 counts as much as two lenses with weight 0.25, and a lens with weight 0 drops out entirely (anything0 = 1). The “soft-power-leaning mix” preset uses weights (0.1, 0.2, 0.2, 0.5). For the same U.S. shares as Ex 2.1:
Compare 16.5 on equal weights: weighting the network lens up, where America is strong, raises its index by seven points. Same data, different theory of power — which is why the sliders exist. Go here for more.
Eq. (2.5) shades, for each empire at each moment, the interval from its harshest lens to its friendliest:
U.S., 2026: shares (4.1, 14.7, 37, 33) → band [4.1, 37] — thirty-three points wide.
A narrow band means every theory of power agrees; a wide one means the verdict is contested by construction. The idea is borrowed from ensemble weather forecasting, where the spread between models is the uncertainty estimate. Go here for more.