▁geomet rical ▁and ▁topological ▁value ▁in ▁bro ader ▁areas . ▁The ▁specific ▁needs ▁of ▁enumer ative ▁geometry ▁were ▁not ▁addressed ▁until ▁some ▁further ▁attention ▁was ▁paid ▁to ▁them ▁in ▁the ▁ 1 9 6 0 s ▁and ▁ 1 9 7 0 s ▁( as ▁pointed ▁out ▁for ▁example ▁by ▁Steven ▁Kle iman ). ▁Inter section ▁numbers ▁had ▁been ▁rig or ously ▁defined ▁( by ▁André ▁We il ▁as ▁part ▁of ▁his ▁found ational ▁programme ▁ 1 9 4 2 – 6 , ▁and ▁again ▁subsequently ), ▁but ▁this ▁did ▁not ▁exhaust ▁the ▁proper ▁domain ▁of ▁enumer ative ▁questions . ▁ ▁F ud ge ▁factors ▁and ▁Hil bert ' s ▁fif teenth ▁problem ▁Na ï ve ▁application ▁of ▁dimension ▁counting ▁and ▁Bé z out ' s ▁theorem ▁yields ▁incorrect ▁results , ▁as ▁the ▁following ▁example ▁shows . ▁In ▁response ▁to ▁these ▁problems , ▁algebraic ▁geom eters ▁introduced ▁vague ▁" f ud ge ▁factors ", ▁which ▁were ▁only ▁rig or ously ▁just ified ▁dec ades ▁later . ▁ ▁As ▁an ▁example , ▁count ▁the ▁con ic ▁sections ▁tang ent ▁to ▁five ▁given ▁lines ▁in ▁the ▁project ive ▁plane . ▁The ▁con ics ▁const itute ▁a ▁project ive ▁space ▁of ▁dimension ▁ 5 , ▁taking ▁their ▁six ▁coefficients ▁as ▁hom ogeneous ▁coordinates , ▁and ▁five ▁points ▁determine ▁a ▁con ic , ▁if ▁the ▁points ▁are ▁in ▁general ▁linear ▁position , ▁as ▁passing ▁through ▁a ▁given ▁point ▁impos es ▁a ▁linear ▁condition . ▁Similarly , ▁tang ency ▁to ▁a ▁given ▁line ▁L ▁( t ang ency