John Calvin on Numbers 29:7-11

7. And ye shall have on the tenth day. This was the day of Atonement. For although they never came into God’s presence without supplication for pardon, they then in a special manner confessed their sins, because a fast was appointed in token of their guilt. For thus it is written in Leviticus 23:29,

"Whatsoever soul it be that shall not be afflicted in that same day, he shall be cut off from among his people."

As to the sacrifices, one bullock only is required; the rest is as before, except that an exception is added, which was omitted in the former cases. For another propitiation is appointed besides the goat, to accord with the fact of their affliction. For the acknowledgment of guilt would have been a dreadful torment to their consciences without the hope of reconciliation. The reason of this sacrifice will be soon explained.