We describe a method to try to construct non-Desarguesian projective planes of a given finite order using a computer program. If a projective plane of order $n$ exists, then it can be constructed from a dual linear space by a sequence of one-line extensions. This motivates a careful analysis of the failures of Desargues' Law in a dual linear space. Up to isomorphism, there are 105 dual linear spaces generated by a non-Desarguesian configuration with which to start the extension process. A further reduction allows us to limit the number of starting configurations to 15.