1 Although the arrangement of the Psalms, which seems to me to contain the secret of a mighty mystery, has not yet been revealed unto me, yet, by the fact that they in all amount to one hundred and fifty, they suggest somewhat even to us, who have not as yet pierced with the eye of our mind the depth of their entire arrangement, whereon we may without being over-bold, so far as God gives, be able to speak. Firstly, the number fifteen, whereof it is a multiple this number fifteen, I say, signifies the agreement of the two Testaments.
For in the former is observed the Sabbath, which signifies rest; in the latter the Lord's Day, which signifies resurrection. The Sabbath is the seventh day, but the Lord's Day, coming after the seventh, must needs be the eighth, and is also to be reckoned the first. For it is called the first day of the week, and so from it are reckoned the second, third, fourth, and so on to the seventh day of the week, which is the Sabbath. But from Lord's Day to Lord's Day is eight days, wherein is declared the revelation of the New Testament, which in the Old was as it were veiled under earthly promises.
Further, seven and eight make fifteen. Of the same number too are the Psalms which are called “of the steps,” because that was the number of the steps of the Temple. Further too, the number fifty in itself also contains a great mystery. For it consists of a week of weeks, with the addition of one as an eighth to complete the number of fifty. For seven times seven make forty-nine, whereto one is added to make fifty. And this number fifty is of so great meaning, that it was after the completion of that number of days from the Lord's Resurrection, that, on the fiftieth day exactly, the Holy Spirit came upon those who were gathered together in Christ.
And this Holy Spirit is in Scripture especially spoken of by the number seven, whether in Isaiah or in the Apocalypse, where the seven Spirits of God are most directly mentioned, on account of the sevenfold operation of one and the self-same Spirit. And this sevenfold operation is mentioned in Isaiah....Hence also the Holy Spirit is spoken of under the number seven. But this period of fifty the Lord divided into forty and ten: for on the fortieth day after His Resurrection He ascended into heaven, and then after ten days were completed He sent the Holy Spirit: under the number forty setting forth to us the period of temporal sojourn in this world.
For the number four prevails in forty; and the world and the year have each four parts; and by the addition of the number ten, as a sort of reward added for the fulfilment of the law in good works, eternity itself is figured. This fifty the number one hundred and fifty contains three times, as though it were multiplied by the Trinity. Wherefore for this reason too we make out that this number of the Psalm is not unsuitable.
Source: The Enarrations, or Expositions, on the Psalms (New Advent)