What did 19th century Presbyterians think about revivals of religion?

William Buell Sprague's 1832 work, Lectures on Revivals of Religion, gives us the answer to this question. His lectures cover an array of topics (the nature of revival, obstacles to revival, divine agency in revival, etc.), and he also includes letters from twenty different Presbyterian clergymen concerning revival. 

Here is a portion of Archibald Alexander's letter, on the nature of true revival:

But I come now to speak of genuine revivals, where the gospel is preached in its purity, and where the people have been well instructed in the doctrines of Christianity. In a revival, it makes the greatest difference in the world whether the people have been carefully taught by catechising, and where they are ignorant of the truths of the Bible. In some cases revivals are so remarkably pure, that nothing occurs with which any pious man can find fault. There is not only no wildness and extravagance, but very little strong commotion of the animal feelings. The word of God distils upon the mind like the gentle rain, and the Holy Spirit comes down like the dew, diffusing a blessed influence on all around. Such a revival affords the most beautiful sight ever seen upon earth. Its aspect gives us a lively idea of what will be the general state of things in the latter-day glory, and some faint image of the heavenly state.

The impressions on the minds of the people in such a work are the exact counterpart of the truth; just as the impression on the wax corresponds to the seal. In such revivals there is great solemnity and silence. The convictions of sin are deep and humbling: the justice of God in the condemnation of the sinner is felt and acknowledged; every other refuge but Christ is abandoned; the heart at first is made to feel its own impenetrable hardness; but when least expected, it dissolves under a grateful sense of God's goodness, and Christ's love; light breaks in upon the soul either by a gradual da^vning, or by a sudden flash; Christ is revealed through the gospel, and a firm and often a joyful confidence of salvation through Him is produced: a benevolent, forgiving, meek, humble and contrite spirit predominates — the love of God is shed abroad—and with some, joy unspeakable and full of glory, fills the soul. A spirit of devotion is enkindled. The word of God becomes exceedingly precious. Prayer is the exercise in which the soul seems to be in its proper element, because by it, God is approached, and his presence felt, and beauty seen: and the new-born soul lives by breathing after the knowledge of God, after communion with God, and after conformity to his will. Now also springs up in the soul an inextinguishable desire to promote the glory of God, and to bring all men to the knowledge of the truth, and by that means to the possession of eternal life. The sincere language of the heart is, "Lord what wouldst thou have me to do?" That God may send upon his church many such revivals, is my daily prayer; and many such have been experienced in our country, and I trust are still going forward in our churches.