In Pursuit of the Unknown Read Free Page B

Locorum Describendorum Ratione (‘Booklet Concerning aWay of Describing Places’). Word of the method spread across Europe, reaching the ears of the Danish nobleman and astronomer Tycho Brahe. In 1579 Tycho used it to make an accurate map of Hven, the island where his observatory was located. By 1615 the Dutch mathematician Willebrord Snellius (Snel van Royen) had developed the method into essentially its modern form: triangulation . The area being surveyed is covered with a network of triangles. By measuring one initial length very carefully, and many angles, the locations of the corners of the triangle, and hence any interesting features within them, can be calculated. Snellius worked out the distance between two Dutch towns, Alkmaar and Bergen op Zoom, using a network of 33 triangles. He chose these towns because they lay on the same line of longitude and were exactly one degree of arc apart. Knowing the distance between them, he could work out the size of the Earth, which he published in his Eratosthenes Batavus (‘The Dutch Eratosthenes’) in 1617. His result is accurate to within 4%. He also modified the equations of trigonometry to reflect the spherical nature of the Earth’s surface, an important step towards effective navigation.
    Triangulation is an indirect method for calculating distances using angles. When surveying a stretch of land, be it a building site or a country, the main practical consideration is that it is much easier to measure angles than it is to measure distances. Triangulation lets us measure a few distances and lots of angles; then everything else follows from the trigonometric equations. The method begins by setting out one line between two points, called the baseline, and measuring its length directly to very high accuracy. Then we choose a prominent point in the landscape that is visible from both ends of the baseline, and measure the angle from the baseline to that point, at both ends of the baseline. Now we have a triangle, and we know one side of it and two angles, which fix its shape and size. We can then use trigonometry to work out the other two sides.
    In effect, we now have two more baselines: the newly calculated sides of the triangle. From those, we can measure angles to other, more distant points. Continue this process to create a network of triangles that covers the area being surveyed. Within each triangle, observe the angles to all noteworthy features – church towers, crossroads, and so on. The same trigonometric trick pinpoints their precise locations. As a final twist, the accuracy of the entire survey can be checked by measuring one of the final sides directly.
    By the late eighteenth century, triangulation was being employed routinely in surveys. The Ordnance Survey of Great Britain began in 1783, taking 70 years to complete the task. The Great Trigonometric Survey ofIndia, which among other things mapped the Himalayas and determined the height of Mount Everest, started in 1801. In the twenty-first century, most large-scale surveying is done using satellite photographs and GPS (the Global Positioning System). Explicit triangulation is no longer employed. But it is still there, behind the scenes, in the methods used to deduce locations from the satellite data.
    Pythagoras’s theorem was also vital to the invention of coordinate geometry. This is a way to represent geometric figures in terms of numbers, using a system of lines, known as axes, labelled with numbers. The most familiar version is known as Cartesian coordinates in the plane, in honour of the French mathematician and philosopher René Descartes, who was one of the great pioneers in this area – though not the first. Draw two lines: a horizontal one labelled x and a vertical one labelled y . These lines are known as axes (plural of axis), and they cross at a point called the origin. Mark points along these two axes according to their distance from the origin, like the markings on a