After examining the whys and wherefores of applied geodesy in high energy particle accelerator construction, the authors show how the transverse beam size and hence the dimension of the magnets are related to positional tolerances and review the methods and instruments developed for this purpose. Three accelerators have successively been built at CERN, the 28 GeV Proton Synchrotron (PS), The Intersecting Storage Rings (ISR), the Super Proton Synchrotron (SPS), and a fourth one is under construction, the Large Electron Positron Collider (LEP). Since 1954, date of the PS construction, many difficulties have been encountered and the authors show how applied geodesy has been able to overcome them. From the traditional geodetic methods available in the 1950's, used to build a 200-m diameter accelerator, up to the 1980's, many methods and instruments ranging from invar wire to Terrameter, have been developed. These have improved the reliability, accuracy and speed of the geodetic measurements and have given a philosophy and a strategy for the Applied Geodesy of a 27-km circumference collider such as LEP.