Appendix — the math, just in time

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.

↩ back to chapter 1

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.

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.

↩ back to chapter 1

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.

↩ back to chapter 1

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):

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.

↩ back to chapter 2

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.

↩ back to chapter 2

Eq. (2.5) shades, for each empire at each moment, the interval from its harshest lens to its friendliest:

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.

↩ back to chapter 2

Eq. (3.3) says the velocity a rocket gains equals its exhaust velocity times the logarithm of its mass ratio. Take a cordite stage with exhaust velocity 2.06 km/s that is two-thirds propellant: it ignites at 3 t and burns out at 1 t:

Note what is absent: the rocket’s size. A 3-gram and a 3,000-tonne rocket with the same mass ratio and exhaust gain exactly the same velocity — which is why no amount of scaling-up rescues a weak propellant, and why the logarithm is called a tyranny: to double the Δv you must square the mass ratio, 3 → 9. Go here for more.

↩ back to chapter 3

Engineers quote propellant performance as “specific impulse” in seconds: how long one kilogram of propellant can push with one kilogram-force of thrust. Multiply by standard gravity, $g_0$ = 9.81 m/s², and you get the exhaust velocity of eq. (3.3):

The chapter’s whole argument sits in that middle line: smokeless powder tripled the number that lives in the rocket equation’s exponent. Go here for more.

↩ back to chapter 3

Eqs. (3.4)–(3.5) for the Victorian stack: five cordite stages, exhaust velocity 2.06 km/s, structure ε = 0.18, each stage delivering 9.4 / 5 = 1.88 km/s:

Each stage keeps 27% of its ignition mass as “payload” — the entire rest of the rocket above it — and five such factors compound to a seven-thousandth. Staging converts an impossible single mass ratio (e^9.4/2.06 ≈ 96) into five easy ones (e^1.88/2.06 ≈ 2.5 each). Go here for more.

↩ back to chapter 3

Orbital speed at 200 km is 7.8 km/s, but eq. (3.2) charges ~9.4. The difference is mostly gravity loss: while the rocket climbs, gravity subtracts $g$ from its acceleration every second, so thrust spent during the climb is partly spent hovering. A rocket that spends ~150 s ascending steeply loses roughly

(plus ~0.1–0.3 km/s of air drag). That is why rockets pitch over toward the horizontal as early as the air allows — every second spent pointing up is rent paid to gravity — and why a real trajectory is a “gravity turn.” Go here for more.

↩ back to chapter 3

Eq. (4.1) makes a technology buildable when every prerequisite group has some arrived member. Bessemer steel needs [coke iron] AND [Watt engine OR waterpower] — the converter must be blown, but anything that blows will do:

Delete Watt and the gate is unchanged — waterpower carries the group. That is why "delete the steam engines" costs so little in Figure 2's world: most of steam's edges sit inside OR-groups. The min picks the earliest substitute; the max insists on every group. Go here for more.

↩ back to chapter 4

Eq. (4.2) turns "possible" into "done." Rocket candy's ingredients were all in place by ~1000 CE and it was first mixed in 1943, so its calibrated lag is roughly 940 years — the lag is the vision desert. Scenario C's program multiplies vision × customer = 6 × 3 = 18 from 1250:

The jitter e^σZ with σ = 0.14 means a typical rollout's lag varies by ~±15%; multiplied down a 43-node chain this is where the wide orbit bands come from. Go here for more.

↩ back to chapter 4

Eq. (4.4) measures a technology's importance by its absence, the same trick neuroscience and machine learning use (lesion studies). Delete "steam engines (all)" from scenario A and the de Laval nozzle loses its historical parent; its OR-group falls back to the next member, castable composites (1942):

A node with no OR-siblings anywhere on the path (radio, printing) returns "never" — which means "no path in this graph," a statement about the model as much as the world. Go here for more.

↩ back to chapter 4