So first, in order to survey the land we created in 1826, the surveying theodolite which is an insturment used to measure vertical and horizontal angles. Together with accurate measurements of short distances, surveyors could use the mathematics of triangles to measure much longer distances and plot geographical features across entire countries. This practice is also called « triangulation ».
The theodolite was used by Britain's government in its first accurate survey of Ireland. Britain had long used surveys to assert authority as its empire grew. Futhermore, theodolites are widely used in surveying today, whether in large-scale map-making, by armed forces in hostile territory, or on building sites and civil engineering projects. Modern total station theodolites measure distances as well as angles, and contain built in recording devices for the results.

Then, in 1663, early surveyors made others surveyor's angle-measuring instrument but still used geometry with triangles because surveving is all about triangles. Indeed, in 1664, so that calculations involving triangles could be simplified, a machine named the trigonometry instrument was used by surveyors. It used a mathematical technique known as trigonometry and it enabled huge areas to be surveyed using the lengths and angles of triangles thanks too three rules which could be moved using dials to make a triangle of the required size where angles or lengths could then be read off directly.

So, to conclude, surveying is one of the oldest mathematical practices, sharing features with navigation, gunnery and civil engineering. The mathematics involves measuring triangles. The results could mean the rise or fall of governments or even empires. And whatever is being surveyed - a building site, battlefield or entire country - the surveyor is building up a series of measurements of angles and lengths in triangles. Surveying remains a specialised and vital mathematical practice. Technologies such as satellites and laser ranging offer powerful new opportunities to increase the speed and precision of surveys.