ThisVolumeisthefirstpartofaworkdesignedtoprovideaconvenientaccountofthefoundationsandmethodsofmodernalgebraicgeometry.Sincenearlyeverytopicofalgebraicgeometryhassomeclaimforinclusionithasbeennecessary￡inordertokeepthesizeofthisvolumewithinreasonablelimits￡toconfineourselvesstrictlytogeneralmethods￡andtostopshortofanydetaileddevelopmentofgeometricalproperties.Wehavethoughtitde8irabletobeginwithasectiondevotedtopurealgebra￡sincethenecessaryalgebraictopicsarenoteasilyaccessibleinEngl18htexts.Afterapreliminarychapteronthebasicnotionsofalgebra￡wedevelopthetheoryofmatrices.Somenoveltyhasbeengiventothisworkbythefactthatthegroundfieldisnotassumedtobecommutative.ThemoregeneralresultsobtainedareusedinChaptersVandVItoanalysetheconceptsonwhichprojectivegeometryisbased.ChaptersIIIandIV￡whichwillberequiredinalatervolume￡aredevotedtoastudyofalgebraicequations.BookIIisconcernedwiththedefinitionandbasicpropertiesofprojectivespaceofndimensions.Boththealgebraicandthe8yntheticdefinitionsarediscussed￡andthetheoryofmatricesoveranon-commutativefieldisusedtoshowthataspacebasedonthepropositionsofincidencecanberepresentedbycoordinates￡withouttheintroductionofanyassumptionequivalenttoPappus\'theorem.Thenecessityofconsideringalargenumberofspecialcase8hasmadeChapterVIratherlong￡butsomespacehasbeen8avedinthelaterpartsofthechapterbymerelymentioningthespecialcasesandleavingtheproofstothereader￡whentheyaresufficientlysimple.Itishopedthatthiswillnotcauseanydifficulty.ThisBookconcludeswithapurelyalgebraicaccountofcollineationsandcorrelations.Certainelementarygeometricalconsequencesareindicated￡butacompletestudyofthegeometricalproblemsinvolvedwouldhavetakenusbeyondourpresentobjective.