A little further in the same book (P. 297 in mine):

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Eighthly, and lastly, A well-grounded assurance sometimes springs from the testimony and witness of the Spirit of God. The Spirit sometimes witnesses to a believer's spirit that he is born of God, that he is beloved of God, that he hath union and communion with God, and that he shall reign for ever with God: Rom. 8.16, "The Spirit itself beareth witness with our spirit, that we are the children of God." The Spirit itself witnesseth not only the gifts and graces of the Spirit, but the Spirit itself witnesseth together with our own spirit, that we are the children of God. Sometimes the saints have two witnesses joining their testimonies together to confirm and establish them in these blessed and glorious truths, that they are the sons of God and heirs of glory; and this is their honour as well as their comfort, that the blessed Spirit should bear witness at the bar of their consciences that they are the sons of God: 1 Cor. 2:12, "Now we have received, not the spirit of the world, but the Spirit which is of God"; that is, that we may know our election, vocation, justification, sanctification, and glorification. A man may receive many things that are freely given of God, and yet not know them till the Spirit comes and makes them known to the soul.

You write:
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Shall we continue to discuss "personal,private revelation" from the Spirit? Or shall we admit that the Spirit speaks through Holy Scripture?

The Spirit makes doctrines of scripture known to the elect, that He does not make known to the unsaved. He also makes some doctrines known to some of the saved and not to other of the saved yet, hence all the great arguments about what scripture actually says. This is called personal, private revelation, much deeper than mere words. If you claim that assurance is ALL and ONLY just because YOU believe scripture, it becomes a little too Arminian at that point. Who is it that gives understanding? The Spirit of God. How does He do it? Personally, and privately. What is understanding? A revealing of the will of God. Hence we (I do anyway) have personal, private revelation by the Spirit of God in regards to my assurance.

Remember:
"Who maketh thee to differ?"