We designate as an empire a state that stands out by area and population, as compared to most neighbors. The same population–technology interaction that enables world population to grow also enables states and then empires to form and expand, mostly by brutal force. Empires form where people are. The joint population share of the top five empires reached 50% of the world population by the year +1, and it did so on just 10% of the dry land area. The areas of top empires expanded in three phases, dependent on message speeds but also skills in delegation of power: Runner, Rider and Engineer Empires. The Rider age produced a new type of nomadic “area empires,” with low population density, in contrast with standard high-density “people empires.” The areas and populations of people empires tend to follow a square root relationship: Their share of world population is the square root of their share of dry land area.

Steamships removed the message speed limit imposed by horses and sails, and telegraphy made communication almost instantaneous. Top state sizes expanded accordingly. Graphs superimpose the growth–decline curves of major post−1800 Engineer Empires. Britain became the largest empire ever (24% of world dry land area), but it lasted at more than half of its maximum size only for 110 years, comparable to nomad Xiongnu. State collapse in China also made Britain briefly the most populous of the world, due to its control of India. For most of the Engineer period Russia has been the largest and China (Qing and People’s Republic) the most populous. India’s population surpassed China’s in 2023. At the 1925 peak of European domination, 64% of Earth’s dry land area was ruled from Europe. It is now down to 21%, mainly Siberia. But European-stock Russia, USA, Canada, Brazil, and Australia remain part of the top seven, along with China and India. Population proportions differ. Since 1800, six to ten states have held more than 2% of Earth’s dry land area. Every half-century, three to four have entered or exited this category. By this pattern, 2000−2050 has been unusually quiet, up to now.

The size of states, including empires, is measured by two crude measures: their area and their population. Areas graphed over time represent the growth–decline curves of empires. These are the basic portraits of empires, showing at a glance how rapidly they formed, how large they became, and how long they lasted. Measuring areas in historical atlases supplies most data. Rome, Ottoman, and Manchu Qing offer examples of completed life cycles. Russia and the USA show curves still incomplete. The beginnings of these curves often approximate the “simple logistic” pattern typical of growth of bacteria colonies when they have plenty of food but space is limited. A table lists the following characteristic of 98 major empires during the last 5000 years: maximum size, duration at least at one-half of maximum size, and rise time from 20−80% of maximum size. Of these, 20 are Runner Empires (3000 to 600 BCE), 66 are Rider Empires (600 BCE to 1800 CE), and 12 are Engineer Empires (from 1800 on).