A method for inhibiting corrosion in industrial cooling waters which contain hardness and a pH of at least 6.5, by dosing the water with a composition which comprises a water-soluble inorganic phosphate capable of inhibiting corrosion in an aqueous alkaline environment and a hydrocarbon polymer containing an N-substituted acrylamide polymers with an amide structure as follows: ##STR1## where R.sub.2 is hydrogen or methyl, where R.sub.1 is a hydrogen or an alkyl and R is alkylene or phenylene, and X is sulfonate, (poly)hydroxyl, (poly)carboxyl or carbonyl, and combinations thereof; or containing derivatized maleic anhydride homo-, co- and terpolymers having N-substituted maleamic acid units, N-substituted maleimide units and maleic acid (and salts) units having a structure as follows: ##STR2## where R.sub.1, R.sub.2 and R.sub.3 are each independently chosen from the group consisting of hydrogen, hydroxyl, carboxyalkyl, carboxyamide, phenyl, substituted phenyl, linear or branched alkyl of from one to ten carbon atoms,and substituted alkyl of from one to ten carbon atoms, where the substituent is phosphonic acid; phosphinic acid; phosphate ester; sulfonic acid; sulfate ester, carboxyamide, (poly)carboxy and (poly)hydroxy, alkoxy and carboxylate ester groups; and combinations thereof.