The present fourth volume recalls Hilbert's axioms from the Foundations of Geometry, and elaborates the theory of geometric constructions. Both extensions and restrictions of the classical Euclidean tools, straightedge and compass, are explained and investigated. One finds constructions by a marked ruler, which allows the duplication of the cube and angle trisection, on the other hand proofs for the impossibility of such constructions by the Euclidean tools or Hilbert tools. The connection to modern algebra, and especially Galois theory is explained.