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The History of Madrid
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How GPS Knows Where You Are
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The Brainless Blob That Redesigned Tokyo’s Railways
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Grab your coffee and let’s dive in.

No law, decree or royal proclamation has ever made Madrid the capital of Spain. In 1561, King Philip II simply moved his court there without any formal announcement, and the government of a vast empire settled into a small Castilian town. Madrid had begun life in the second half of the ninth century as a fortress built for Muhammad I, the emir of Córdoba, whose Umayyad dynasty ruled most of Muslim Spain. Toledo, a large city in the middle of the peninsula, kept rebelling against Umayyad rule, so the emir planted his fortress on a steep bank above the Manzanares River to stop help from reaching the rebels. A small settlement called Mayrit, a name usually linked to the water beneath the site, grew up around its walls. Even the most generous estimate puts it at about 2,000 people in the tenth century, squeezed onto nine hectares.

Castile, the Christian kingdom pushing south, took the fortress under its king, Alfonso VI, in 1083 or 1085, and historians still argue over which. In one version the town was handed over by negotiation before Toledo itself fell in 1085, and in the other it came afterward. For nearly five centuries Madrid stayed a middle-ranking Castilian town, while the old Muslim fortress was rebuilt bit by bit into a royal palace called the Alcázar. At most 20,000 people lived there when Philip II brought the court in. Courtiers, officials and servants poured in behind him, and by the end of the century the population was approaching 100,000.

In 1601, the Duke of Lerma, the favorite and closest adviser of Philip III, persuaded the king to move the court north to Valladolid, where Lerma had been quietly buying property. Around half of Madrid, somewhere between 50,000 and 60,000 people, followed it, and house prices in the city collapsed. Lerma then bought cheap property in Madrid. Before long he was suggesting to the king that the court might be better off back where it had been, and the city offered 250,000 ducats toward the cost of the move. The king accepted. In 1606, after five years in Valladolid, the court came back to Madrid, where Lerma was now a landlord too. The Alcázar survived until Christmas Eve of 1734, when a fire broke out in the rooms of Jean Ranc, the French painter to the royal court, and burned for four days, leaving the old palace in ruins. The Royal Palace built on the same spot between 1738 and 1755 is still the official residence of the Spanish royal family.

The US Space Force promises to keep at least 24 GPS satellites working 95 percent of the time, and it currently runs 31 of them, each circling the Earth twice a day about 20,200 kilometers up. Every one carries atomic clocks and broadcasts, over and over, where it is and exactly what time it is. Your phone notes when each signal arrives, works out how long it spent traveling, and turns that into a distance. Three distances are enough to pin down a position, a technique called trilateration. A fourth satellite lets the phone correct its own clock, which is nowhere near as good as the ones in orbit. Light covers about 30 centimeters in a billionth of a second, so a clock that is 10 billionths of a second out puts you about 3 meters from where you really are. For GPS to work, the satellite clocks have to stay accurate to within 20 to 30 billionths of a second.

Under Albert Einstein's special theory of relativity, a moving clock runs slow, and the satellites travel fast enough to lose 7 millionths of a second a day against clocks on the ground. Under his general theory, clocks run faster where gravity is weaker, and at 20,200 kilometers up they gain 45 millionths of a second a day. The two effects pull in opposite directions but do not cancel out. Together they leave each satellite clock running 38 millionths of a second fast every day, more than a thousand times the error the system can tolerate. Left uncorrected, according to Richard Pogge, an astronomer at Ohio State University, positions would start drifting within about two minutes and the error would grow by around 10 kilometers a day.

Engineers deal with most of the problem before the satellites ever leave the ground. The satellite clocks are set to tick slightly slower than identical clocks on the ground, so that once they reach orbit, relativity speeds them up to the right rate. The receiver in your phone handles the remaining corrections, using data the satellites send down along with their position and time. Without those corrections, Pogge writes, the system would be "utterly worthless for navigation in a very short time." GPS began as a military project, declared a strategic system by the US Department of Defense in 1978. Today the signal is free for anyone to use, from taxi drivers and delivery riders to hikers who have wandered off the trail.