A graph superimposes the growth–decline curves of major Stirrup Rider Empires, from 600 to 1200. While being a major advance in horse riding, the stirrup just offers a short term for the intermediary phase of Rider Empires. Expansion of Islamic Caliphate was the towering event. It surpassed the Xiongnu area record. Apart from Tang, Tufan in Tibet, Liao, and Seljuk, all other medieval empires remained of modest size. In population, Song in 1125 briefly reached 38% of the world population – the largest percentage any empire has ever reached. The Caliphate altered the language mix throughout North Africa, introducing the Arabic. The Seljuks did so from Central Asia to Anatolia, introducing Turkic. The Caliphate clashed with Tang in Central Asia in 751. The forces of an empire reaching to the Atlantic Ocean confronted for the first time those of an empire that reached to the Pacific. Neither realized the momentousness of this skirmish. Western Europe developed feudalism, a maddeningly complex multistranded hierarchical order, which does away with single territorial authority.

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.

A graph superimposes the growth–decline curves of major Early Rider Empires, from 600 BCE to 600 CE. Compared to running messengers, the Achaemenid empire sharply boosted message speed by using horse relay stations. A major rise in empire area resulted, also helped by split delegation of provincial power. The Xiongnu steppe empire broke the Achaemenid size record hugely but briefly, while Han did so longer but minimally. Rome fell short in size but lasted longer. By population size Han mostly was top, but Maurya and Rome at times surpassed it. Most of these empires profited from religious tolerance, compared to previous theocracies; Sassanids were an exception. The Qin-Han rule established a harsh centralized bureaucracy, while Rome limited it, allowing for local autonomy. Han official contact with Hellenic Bactria in 127 BCE marks the first indirect linkage of states from the Pacific to the Atlantic.

A graph superimposes the growth–decline curves of major Late Rider Empires, from 1200 to 1800. This was a period of shift from horses to sails. West European feudalism indirectly led to respect for law and a curiosity revolution. Inciting exploration, the latter began to give rise to transoceanic empires, first of all Spanish. But at first, Mongol horsemen seized a record-size area, unsurpassed among land-borne empires. Up to 1750 all truly large empires except Spanish remained land bound: Manchu Qing, Russia, Ottoman, and Mughal. The burgeoning French, Portuguese, and British empires were still modest. Much of the Mongol hold meant sparsely populated superficial tribute area, but by conquering China the Mongol empire also became the world’s most populous. Later on, Ming, Mughal, and Qing shared this eminence. Nomad empires were a phenomenon that rose and ended with the Rider phase. The Inca and Aztec empires retraced from scratch the human self-domestication process that the Old World underwent thirty-five centuries earlier, but they still lacked metals and the wheel.

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).