Psalm 50:15: "'and call upon me in the day of trouble; I will deliver you, and you shall glorify me.'"
Philippians 4:6: "do not be anxious about anything, but in everything by prayer and supplication with thanksgiving let your requests be made known to God."
Isaiah 65:24: "Before they call I will answer; while they are yet speaking I will hear."
1 John 5:14: "And this is the confidence that we have toward him, that if we ask anything according to his will he hears us."
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Psalm 50:15 and Philippians 4:6 seem to appear in the lines "In holy conversation / We speak to God in prayer" at the beginning of the first verse, although they also inform the hymn generally. Isaiah 65:24 and 1 John 5:14 are alluded to in the lines "With care our Father listens / To ev'ry thought expressed, / Then answers our petitions / In ways He knows are best" at the end of the second verse and (particularly 1 John 5:14) in the lines "So let us pray securely, / Expressing hopes and fears / With confidence that surely / Our Father ever hears" at the end of the third.
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The lines "And to support our prayer, / His Spirit intercedes" at the end of the first verse seem to be drawn from Romans 8:26: "Likewise the Spirit helps us in our weakness. For we do not know what to pray for as we ought, but the Spirit himself intercedes for us with groanings too deep for words."
As with the previous hymn, the lines "Then [our Father] answers our petitions / In ways He knows are best" may refer to Romans 8:28 ("And we know that for those who love God all things work together for good, for those who are called according to his purpose.") or even Genesis 50:20, where Joseph tells his brothers, "'As for you, you meant evil against me, but God meant it for good, to bring it about that many people should be kept alive, as they are today.'"