Triangulation - History

History

Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and gun direction of weapons.

The use of triangles to estimate distances goes back to antiquity. In the 6th century BC the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids by measuring the length of their shadows and that of his own at the same moment, and comparing the ratios to his height (intercept theorem); and to have estimated the distances to ships at sea as seen from a clifftop, by measuring the horizontal distance traversed by the line-of-sight for a known fall, and scaling up to the height of the whole cliff. Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope, i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra, the forerunner of the Arabic alidade. A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria (c. 10–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe. In China, Pei Xiu (224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establish distances; while Liu Hui (c. 263) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places.

In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the agromensores; 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. Increased awareness and use of such techniques in Spain may be attested by the medieval Jacob's staff, used specifically for measuring angles, which dates from about 1300; and the appearance of accurately surveyed coastlines in the Portolan charts, the earliest of which that survives is dated 1296.