06/04/2018

Mathematics is behind GPS and modern navigation.

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

The position of people and objects on the Earth's surface is given by two numbers, both measured in degrees. Latitude, which ranges from 90° north (at the North Pole) to 90° south (at the South Pole), describes the position relative to the equator. Longitude, which ranges from 180° east to 180° west, indicates the position relative to a certain reference meridian.

The choice of this meridian is arbitrary. Currently we use the one from the Greenwich Observatory in London, but the first person to represent lines of latitude and longitude on a map, the Greco-Roman mathematician and astronomer Ptolemy (2nd century), preferred the meridian of the Happy Islands, the current archipelago of Madeira.

Latitude is easy to calculate from the height, relative to the horizon, of certain stars and constellations (Polaris, Southern Cross) or of the sun itself at midday. Measuring instruments created in antiquity, such as the astrolabe, were improved by the Portuguese and other navigators, so that, by the end of the 16th century, the calculation of longitude had become precise and routine.

Longitude is a much more delicate problem. Although it is possible to determine it from astronomical phenomena, such as eclipses of Jupiter's moons, such observations are very difficult from a moving ship. Thus, longitude continued to be estimated by guesswork, to the serious detriment of navigation.

Of course, many solutions were proposed, some as innovative as they were ineffective. In his book "The Island of the Day Before," the Italian writer Umberto Eco (1932-2016) describes one of the most creative, and most cruel.

A wounded dog was loaded onto the ship. The knife that had caused the injury remained ashore and, every day at noon, was placed in the fire. According to the theories of the English diplomat Sir Kenelm Digby (1603-1665), the weapon and the wound remained linked by a "sympathy": the fire on the knife would instantly cause the dog terrible pain, even from a distance. By the animal's whimpers, the ship's crew would know it was noon at the port of departure, and with this information it would be easy to calculate the ship's longitude. They just couldn't let the wound heal, reopening it whenever necessary. Poor dogs, victims of the superstitions of Sir Kenelm and his followers…

Given the enormous economic interest, it is no surprise that great efforts were devoted to solving the problem of longitude. Seafaring nations, such as Portugal, Spain, and the Netherlands, offered rewards for the solution, but these prizes were never awarded. Later, France would create a research center specifically dedicated to the subject, the Bureau des Longitudes, which still exists today.

In 1714, the English Parliament passed the famous Longitude Act: “the discovery of longitude is of such importance to Great Britain, to the safety of her navy and her merchant fleet, that […] a Longitude Prize shall be established for the person or persons who discover longitude.”

The main winner was the Englishman John Harrison (1693 – 1776), who built a portable clock that worked well even on a ship at sea. From then on, ships could know the time at their port of departure at any moment (without having to torture animals!), which was enough to obtain the longitude.

Harrison achieved another feat: receiving the prize money, albeit only after four decades of intrigue and legal disputes with the authorities.

All of this became obsolete with the advent of GPS (Global Positioning System). Initially created by the United States armed forces for military purposes, GPS made navigation accessible to everyone, at sea, on land, and in the air, becoming one of the most incredible applications of mathematics in everyday life.

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