▁ ▁Hil bert ' s ▁fif teenth ▁problem ▁was ▁to ▁overcome ▁the ▁apparently ▁arbitrary ▁nature ▁of ▁these ▁inter vent ions ; ▁this ▁aspect ▁goes ▁beyond ▁the ▁found ational ▁question ▁of ▁the ▁Sch ub ert ▁calculus ▁itself . ▁ ▁C lem ens ▁conject ure ▁ ▁In ▁ 1 9 8 4 ▁H . ▁C lem ens ▁studied ▁the ▁counting ▁of ▁the ▁number ▁of ▁rational ▁curves ▁on ▁a ▁qu int ic ▁three fold ▁ ▁and ▁reached ▁the ▁following ▁conject ure . ▁ ▁Let ▁ ▁be ▁a ▁general ▁qu int ic ▁three fold , ▁ ▁a ▁positive ▁integer , ▁then ▁there ▁are ▁only ▁a ▁finite ▁number ▁of ▁rational ▁curves ▁with ▁degree ▁ ▁on ▁. ▁ ▁This ▁conject ure ▁has ▁been ▁resolved ▁in ▁the ▁case ▁, ▁but ▁is ▁still ▁open ▁for ▁higher ▁. ▁ ▁In ▁ 1 9 9 1 ▁the ▁paper ▁about ▁mirror ▁symmetry ▁on ▁the ▁qu int ic ▁three fold ▁in ▁ ▁from ▁the ▁string ▁theoretical ▁view point ▁gives ▁numbers ▁of ▁degree ▁d ▁rational ▁curves ▁on ▁ ▁for ▁all ▁. ▁Prior ▁to ▁this , ▁algebraic ▁geom eters ▁could ▁calculate ▁these ▁numbers ▁only ▁for ▁. ▁ ▁Ex amples ▁ ▁Some ▁of ▁the ▁histor ically ▁important ▁examples ▁of ▁enumer ations ▁in ▁algebraic ▁geometry ▁include : ▁▁▁ 2 ▁ ▁The ▁number ▁of ▁lines ▁meeting ▁ 4 ▁general ▁lines ▁in ▁space ▁ 8 ▁▁ ▁The ▁number ▁of ▁circles ▁tang ent ▁to ▁ 3 ▁general ▁circles ▁( the ▁problem ▁of ▁Ap oll on ius ). ▁ 2 7 ▁▁ ▁The ▁number ▁of ▁lines ▁on ▁a ▁smooth ▁cub ic ▁surface ▁(