Biblical Commentary on the Old Testament, by Carl Friedrich Keil and Franz Delitzsh, [1857-78], at sacred-texts.com
The festal sacrifice for the new moon of the seventh month consisted of a burnt-offering of one bullock, one ram, and seven yearling lambs, with the corresponding meat-offerings and drink-offerings, and a sin-offering of a he-goat, "besides" (i.e., in addition to) the monthly and daily burnt-offering, meat-offering, and drink-offering. Consequently the sacrifices presented on the seventh new moon's day were, (1) a yearling lamb in the morning and evening, with their meat-offering and drink-offering; (2) in the morning, after the daily sacrifice, the ordinary new moon's sacrifice, consisting of two bullocks, one ram, and seven yearling lambs, with their corresponding meat-offerings and drink-offerings (see at Num 29:11); (3) the sin-offering of the he-goat, together with the burnt-offering of one bullock, one ram, and seven yearling lambs, with their proper meat-offerings and drink-offerings, the meaning of which has been pointed out at Lev 23:23.
On the day of atonement, on the tenth of the seventh month, a similar festal sacrifice was to be offered to the one presented on the seventh new moon's day (a burnt-offering and sin-offering), in addition to the sin-offering of atonement prescribed at Lev 16, and the daily burnt-offerings. For a more minute description of this festival, see at Lev 16 and Lev 23:26-32.
The feast of Tabernacles, the special regulations for the celebration of which are contained in Lev 23:34-36 and Lev 23:39-43, was distinguished above all the other feasts of the year by the great number of burnt-offerings, which raised it into the greatest festival of joy. On the seven feast-days, the first of which was to be celebrated with sabbatical rest and a holy meeting, there were to be offered, in addition to the daily burnt-offering, every day a he-goat for a sin-offering, and seventy oxen in all for a burnt-offering during the seven days, as well as every day two rams and fourteen yearling lambs, with the requisite meat-offerings and drink-offerings. Whilst, therefore, the number of rams and lambs was double the number offered at the Passover and feast of Pentecost, the number of oxen was fivefold; for, instead of fourteen, there were seventy offered during the seven days. This multiplication of the oxen was distributed in such a way, that instead of there being ten offered every day, there were thirteen on the first day, twelve on the second, and so on, deducting one every day, so that on the seventh day there were exactly seven offered; the arrangement being probably made for the purpose of securing the holy number seven for this last day, and indicating at the same time, through the gradual diminution in the number of sacrificial oxen, the gradual decrease in the festal character of the seven festal days. The reason for this multiplication in the number of burnt-offerings is to be sought for in the nature of the feast itself. Their living in booths had already visibly represented to the people the defence and blessing of their God; and the foliage of these booths pointed out the glorious advantages of the inheritance received from the Lord. But this festival followed the completion of the ingathering of the fruits of the orchard and vineyard, and therefore was still more adapted, on account of the rich harvest of splendid and costly fruits which their inheritance had yielded, and which they were about to enjoy in peace now that the labour of agriculture was over, to fill their hearts with the greatest joy and gratitude towards the Lord and Giver of them all, and to make this festival a speaking representation of the blessedness of the people of God when resting from their labours. This blessedness which the Lord had prepared for His people, was also expressed in the numerous burnt-offerings that were sacrificed on every one of the seven days, and in which the congregation presented itself soul and body to the Lord, upon the basis of a sin-offering, as a living and holy sacrifice, to be more and more sanctified, transformed, and perfected by the fire of His holy love (see my Archol. i. p. 416).
The eighth day was to be azereth, a closing feast, and only belonged to the feast of Tabernacles so far as the Sabbath rest and a holy meeting of the seventh feast-day were transferred to it; whilst, so far as its sacrifices were concerned, it resembled the seventh new moon's day and the day of atonement, and was thus shown to be the octave or close of the second festal circle (see at Lev 23:36).
The sacrifices already mentioned were to be presented to the Lord on the part of the congregation, in addition to the burnt-offerings, meat-offerings, drink-offerings, and peace-offerings which individuals or families might desire to offer either spontaneously or in consequence of vows. On the vowing of burnt-offerings and peace-offerings, see Num 15:3, Num 15:8; Lev 22:18, Lev 22:21.
Num 29:40 forms the conclusion of the list of sacrifices in ch. 28 and 29.