Napoleon’s theorem
The essential data that determines a triangle consists of knowing the length of
one side and two angles. By using trigonometry we can measure everything else.
In surveying areas of land in order to draw maps it is quite useful to be a ‘flat-
earther’ and assume triangles to be flat. A network of triangles is established by
starting with a base line BC of known length, choosing a distant point A (the
triangulation point) and measuring the angles A C and AĈB by theodolite. By
trigonometry everything is known about the triangle ABC and the surveyor
moves on, fixes the next triangulation point from the new base line AB or AC and
repeats the operation to establish a web of triangles. The method has the
advantage of being able to map inhospitable country involving such barriers as
marshland, bogs, quicksand and rivers.
It was used as the basis for the Great Trigonometrical Survey of India which
began in the 1800s and lasted 40 years. The object was to survey and map along
the Great Meridional Arc from Cape Comorin in the south to the Himalayas in the
north, a distance of some 1500 miles. To ensure utmost accuracy in measuring
angles, Sir George Everest arranged the manufacture of two giant theodolites in
London, together weighing one ton and needing teams of a dozen men to
transport them. It was vital to get the angles right. Accuracy in measurement was
paramount and much talked about but it was the humble triangle which was at
the centre of operations. The Victorians had to make do without GPS though they
did have computers – human computers. Once all the lengths in a triangle have
been computed, the calculation of area is straightforward. Once again, the
triangle is the unit. There are several formulae for the area A of a triangle, but
the most remarkable is Heron of Alexandria’s formula:
marcin
(Marcin)
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