When Rabbi Shesheth kept a fast, on concluding his prayer he added the following: Sovereign of the Universe, Thou know full well that in the time when the Temple was standing, if a man sinned he used to bring a sacrifice, and though all that was offered of it was its fat and blood, atonement was made for him therewith. Now I have kept a fast and my fat and blood have diminished. May it be Thy will to account my fat and blood which have been diminished as if I had offered them before Thee on the altar, and do Thou favor me.

(Babylonian Talmud, Berachot 17a)

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[IF ONE SAYS]: I SHALL SIN AND THE DAY OF ATONEMENT WILL PROCURE ATONEMENT FOR ME, THE DAY OF ATONEMENT PROCURES FOR HIM NO ATONEMENT. FOR TRANSGRESSIONS AS BETWEEN MAN AND THE OMNIPRESENT THE DAY OF ATONEMENT PROCURES ATONEMENT, BUT FOR TRANSGRESSIONS AS BETWEEN MAN AND HIS FELLOW THE DAY OF ATONEMENT DOES NOT PROCURE ANY ATONEMENT, UNTIL HE HAS PACIFIED HIS FELLOW. THIS WAS EXPOUNDED BY R. ELEAZAR B. ALARIAH: FROM ALL YOUR SINS BEFORE THE LORD SHALL YE BE CLEAN,13 I.E., FOR TRANSGRESSIONS AS BETWEEN MAN AND THE OMNIPRESENT THE DAY OF ATONEMENT PROCURES ATONEMENT, BUT FOR TRANSGRESSIONS AS BETWEEN MAN AND HIS FELLOW THE DAY OF ATONEMENT DOES NOT PROCURE ATONEMENT UNTIL HE HAS PACIFIED HIS FELLOW.14 R. AKIBA SAID: HAPPY ARE YOU, ISRAEL! WHO IS IT BEFORE WHOM YOU BECOME CLEAN? AND WHO IS IT THAT MAKES YOU CLEAN? YOUR FATHER WHICH IS IN HEAVEN, AS IT IS SAID: AND I WILL SPRINKLE CLEAN WATER UPON YOU AND YE SHALL BE CLEAN.15 AND IT FURTHER SAYS: THOU HOPE OF ISRAEL, THE LORD!16 JUST AS THE FOUNTAIN RENDERS CLEAN THE UNCLEAN, SO DOES THE HOLY ONE, BLESSED BE HE, RENDER CLEAN ISRAEL.

(Babylonian Talmud, Yoma, Chapter 8, Page 85, 2)