Commentaries:
Robertson's Word Pictures (NT)
To be saved (ina swqw). Final clause with ina and first aorist passive subjunctive. What did he mean by "saved"? Certainly more than escape from peril about the prisoners or because of the earthquake, though these had their influences on him. Cf. way of salvation in verse Acts 16:17.
Believe on the Lord Jesus (Pisteuson epi ton kurion Ihsoun). This is what Peter told Cornelius (Acts 10:43). This is the heart of the matter for both the jailor and his house.
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