(), How to measure the height of a sea island. Illustration from an edition of 1726 's 1533 proposal to use triangulation for mapmaking Triangulation today is used for many purposes, including
surveying,
navigation,
metrology,
astrometry,
binocular vision,
model rocketry and gun direction of
weapons. In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the
agrimensores; but were introduced into medieval Spain through
Arabic treatises on the
astrolabe, such as that by
Ibn al-Saffar (d. 1035).
Abu Rayhan Biruni (d. 1048) also introduced triangulation techniques to
measure the size of the Earth and the distances between various places. Simplified Roman techniques then seem to have co-existed with more sophisticated techniques used by professional surveyors. But it was rare for such methods to be
translated into Latin (a manual on geometry, the eleventh century
Geomatria incerti auctoris is a rare exception), and such techniques appear to have percolated only slowly into the rest of Europe. In England Frisius's method was included in the growing number of books on surveying which appeared from the middle of the century onwards, including
William Cuningham's
Cosmographical Glasse (1559), Valentine Leigh's
Treatise of Measuring All Kinds of Lands (1562),
William Bourne's
Rules of Navigation (1571),
Thomas Digges's
Geometrical Practise named Pantometria (1571), and
John Norden's ''Surveyor's Dialogue'' (1607). It has been suggested that
Christopher Saxton may have used rough-and-ready triangulation to place features in his county maps of the 1570s; but others suppose that, having obtained rough bearings to features from key vantage points, he may have estimated the distances to them simply by guesswork.
Willebrord Snell The modern systematic use of triangulation networks stems from the work of the Dutch mathematician
Willebrord Snell, who in 1615 surveyed the distance from
Alkmaar to
Breda, approximately 72 miles (116 kilometres), using a chain of quadrangles containing 33 triangles in all. Snell underestimated the distance by 3.5%. The two towns were separated by one degree on the
meridian, so from his measurement he was able to calculate a value for the circumference of the earth – a feat celebrated in the title of his book
Eratosthenes Batavus (
The Dutch Eratosthenes), published in 1617. Snell calculated how the planar formulae could be corrected to allow for the curvature of the earth. He also showed how to
resection, or calculate, the position of a point inside a triangle using the angles cast between the vertices at the unknown point. These could be measured much more accurately than bearings of the vertices, which depended on a compass. This established the key idea of surveying a large-scale primary network of control points first, and then locating secondary subsidiary points later, within that primary network.
Further developments Snell's methods were taken up by
Jean Picard who in 1669–70 surveyed one degree of latitude along the
Paris Meridian using a chain of thirteen triangles stretching north from
Paris to the clocktower of
Sourdon, near
Amiens. Thanks to improvements in instruments and accuracy, Picard's is rated as the first reasonably accurate measurement of the radius of the earth. Over the next century this work was extended most notably by the Cassini family: between 1683 and 1718
Jean-Dominique Cassini and his son
Jacques Cassini surveyed the whole of the Paris meridian from
Dunkirk to
Perpignan; and between 1733 and 1740 Jacques and his son
César Cassini undertook the first triangulation of the whole country, including a re-surveying of the
meridian arc, leading to the publication in 1745 of the first map of France constructed on rigorous principles. Triangulation methods were by now well established for local mapmaking, but it was only towards the end of the 18th century that other countries began to establish detailed triangulation network surveys to map whole countries. The
Principal Triangulation of Great Britain was begun by the
Ordnance Survey in 1783, though not completed until 1853; and the
Great Trigonometric Survey of India, which ultimately named and mapped
Mount Everest and the other Himalayan peaks, was begun in 1801. For the Napoleonic French state, the French triangulation was extended by
Jean-Joseph Tranchot into the German
Rhineland from 1801, subsequently completed after 1815 by the Prussian general
Karl von Müffling. Meanwhile, the mathematician
Carl Friedrich Gauss was entrusted from 1821 to 1825 with the triangulation of the
kingdom of Hanover (), on which he applied the
method of least squares to find the best fit solution for problems of large systems of
simultaneous equations given more real-world measurements than unknowns. Today, large-scale triangulation networks for positioning have largely been superseded by the
global navigation satellite systems established since the 1980s, but many of the control points for the earlier surveys still survive as valued historical features in the landscape, such as the concrete
triangulation pillars set up for
retriangulation of Great Britain (1936–1962), or the triangulation points set up for the
Struve Geodetic Arc (1816–1855), now scheduled as a UNESCO
World Heritage Site. == See also ==