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Matthew 16:28 Truly, I say to you, there are some standing here who will not taste death before they see the Son of man coming in his kingdom.
Some think that it was to take place at the resurrection, and in
the day of judgment, of which Christ spake in the preceding
verse. But I say it took place in the Transfiguration of
Christ. For in it they beheld Christ’s glorious kingdom as
in a glass. Three of the Apostles, namely, Peter, James,
and John, had a foretaste of this kingdom. This view is
plain from what follows. All the three Evangelists who
relate the Transfiguration, place it immediately after this
promise, as though it were the fulfilment of it. Thus SS.
Hilary, Chrysostom, Jerome, Ambrose, Theophylact, and
others, passim. Whence St. Leo says (de Transfig.). In the
kingdom, that is in royal splendour. For in His Transfiguration
Christ gave to His Apostles a specimen of the glory,
the joy and the happiness which the Saints shall obtain in
the Heavenly Kingdom, that He might thereby animate
them to Evangelical labours and sorrows, and that they
might animate others to the same.
A likely cross-reference (many thanks to Nick in the combox below), to cast further light on this saying of Jesus, is the following:
2 Peter 1:16-18 For we did not follow cleverly devised myths when we made known to you the power and coming of our Lord Jesus Christ, but we were eyewitnesses of his majesty.  For when he received honor and glory from God the Father and the voice was borne to him by the Majestic Glory, “This is my beloved Son, with whom I am well pleased,”  we heard this voice borne from heaven, for we were with him on the holy mountain.
Here’s another similar instance of this sort:
Mark 13:2 And Jesus said to him, “Do you see these great buildings? There will not be left here one stone upon another, that will not be thrown down.”
It is argued by some that the Wailing Wall in Jerusalem (a remnant of Herod’s Temple) disproves this prophecy, since it has many stones on top of each other.
I would say here that such statements need not (and should not) be taken literally at all, since exaggeration or hyperbole to stress a point is very common in Scripture. Literal intent is often not intended in the first place, and this was understood in the culture. Jesus was simply saying that great destruction was to happen in Jerusalem, and even (unthinkably!) that the temple itself would be destroyed. In order to drive the point home dramatically, He said “not one stone would be left on another,” rather than saying, “just one wall would be left.”
Another example would be when He referred to faith being able to move a mountain (Matt 17:20; 21:21; Mk 11:23; cf. St. Paul: 1 Cor 13:2). Relentless literalism in expression and interpretation is far more a function of post-Enlightenment rationalism than ancient Jewish thinking. Unfortunately, we often extrapolate our uniquely modern methods and premises anachronistically back to the Bible. This would be a case of that, in my opinion. So it has to do, I believe, with how descriptive language was used and understood in that culture.
Hyperbole in the Bible is extremely common (and that is no exaggeration!). No one need take my word on that alone. In the book, The Bible as Literature: An Introduction (Oxford University Press, 2nd ed., 1990: another one-cent used book on amazon!), co-authors and English professors John B. Gabel and Charles B. Wheeler provide many examples on pages 22-24:
Can we say that such instances as those above have to do with the Two Natures of Jesus (Divine and Human), and the limitation of the latter? Jesus, for example, appears to state that He did not know the exact day of the final Judgment:
1) Abraham’s descendants will be “countless as the dust of the earth” (Gen 13:16).
2) The same are described as “numerous as the stars in the sky and the grains of sand on the sea-shore” (Gen 22:17).
3) The authors state (pp. 22-23), “That such phrases were conventional devices is shown by their appearance in other contexts, for example, . . .”:
A) Genesis 41:49 And Joseph stored up grain in great abundance, like the sand of the sea, until he ceased to measure it, for it could not be measured.
B) Joshua 11:4 And they came out, with all their troops, a great host, in number like the sand that is upon the seashore, with very many horses and chariots.
C) 1 Samuel 13:5 And the Philistines mustered to fight with Israel, thirty thousand chariots, and six thousand horsemen, and troops like the sand on the seashore in multitude; . . .
D) 2 Samuel 17:11 But my counsel is that all Israel be gathered to you, from Dan to Beer-sheba, as the sand by the sea for multitude, . . .
4) “The adjective ‘all’ is frequently used to express similar conventional hyperbole” (p. 23). The authors give as examples of this, 2 Sam 3:37; 1 Kgs 18:19, and 2 Sam 16:22.
5) Examples of exaggeration in description of military exploits occur in passages such as Judges 6:5: “For they would come up with their cattle and their tents, coming like locusts for number; both they and their camels could not be counted;. . .”
6) “Hyperbole reaches its climax in the narrative portions of Daniel and the entire book of Esther; in both of these, hyperbole is so constant that it can no longer be regarded as merely a device: It must be seen as an intrinsic part of the author’s conception of his subject” (pp. 23-24). Examples given are the furnace heated to seven times its usual heat (Dan 3:19) and a 75-foot high gallows (Est 5:14).
7) Three examples from Jesus Himself are also provided:
A) Matthew 23:23-24 “Woe to you, scribes and Pharisees, hypocrites! for you tithe mint and dill and cummin, and have neglected the weightier matters of the law, justice and mercy and faith; these you ought to have done, without neglecting the others.  You blind guides, straining out a gnat and swallowing a camel!
B) Matthew 19:24 Again I tell you, it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God.
C) Matthew 18:8-9 And if your hand or your foot causes you to sin, cut it off and throw it away; it is better for you to enter life maimed or lame than with two hands or two feet to be thrown into the eternal fire.  And if your eye causes you to sin, pluck it out and throw it away; it is better for you to enter life with one eye than with two eyes to be thrown into the hell of fire.
Matthew 24:36 But of that day and hour no one knows, not even the angels of heaven, nor the Son, but the Father only.
Mark 13:32 But of that day or that hour no one knows, not even the angels in heaven, nor the Son, but only the Father.
I don’t believe that explanation applies to the first two instances, dealt with above, because then it would be a case of His positively uttering errors of fact, which could not happen (since He knew all things in His Divine Nature). I think both are plausibly explained in the above manner: the first having to do with a particular sense of “kingdom” and a manifestation of Jesus’ “coming” in some sort of extraordinary glorified fashion, and the second with Hebrew linguistic conventions.
Not knowing the day or hour, on the other hand, is indeed arguably an instance of His specifically referring to His Human Nature (and even that in one limited, specific sense). But even so, He knew it in His Divine Nature. I would say it is the distinction between stating a falsehood and the limitations of human nature, even for the Son of God (the incarnation being a “lowering” in certain ways: cf. Heb 2:7, 9). He could express the latter because it doesn’t contradict what He knows as God the Son, whereas the former (stating a demonstrable error) would be a contradiction to the Hypostatic Union, since it would go against His knowing all things (as He does) in His Divine Nature. The Two Natures of Jesus preclude His uttering an error.
That’s how I understand it, anyway. I don’t claim to be any sort of expert on fine points of Christology such as this. Another option is expressed by some of the fathers:
St. Jerome: Having then shewn that the Son of God cannot
be ignorant of the day of the consummation, we must
now show a cause why He should be said to be ignorant.
When after the resurrection He is demanded concerning
this day by the Apostles, He answers more openly; “It is
not for you to know the times or the seasons which the
Father has put in his own power.” [Acts 1:7] Wherein He
shews that Himself knows, but that it was not expedient
for the Apostles to know, that being in uncertainty of
the coming of their Judge, they should live every day as
though they were to be judged that day.
St. Augustine: de Trin., i, 12: When He says here,
“Knows not,” He means, ‘makes others not to know;’ i.e.
He knew not then, so as to tell His disciples; as it was said
to Abraham, “Now I know that thou fearest God;” [Gen
22:19] i.e. ‘Now have I caused that thou shouldest know,’
because by the temptation he came to know himself.
St. Augustine: Lib. 83, Quaest. Q60: That the Father alone knows may
be well understood in the above-mentioned manner of knowing, that He
makes the Son to know; but the Son is said not to know, because be
does not make men to know.
Lapide presents aspects of both explanations and adds an additional interesting possible interpretation:
for so it is in the Greek, Latin, Syriac, Arabic, Persian,
Egyptian, Ethiopic. Various answers are given. The best
is that which is common among the Fathers. It is that
the Son, both as God and as man, by infused knowledge,
knows the Day of Judgment and of the end of the world,
for it pertains for Him to know this, inasmuch as He has
been appointed the Judge of the world. But Christ denies
that He knoweth this as man, and as He is God’s messenger
to us, because He did not know it so that He could
reveal it to us, or because He had not been commissioned
by the Father to reveal it to us. As an ambassador who
was questioned concerning the secrets of his prince would
reply that he did not know them, although he did know
them, because he did not know them as an ambassador.
For an ambassador declares only those things which he
has a commission to declare.
Christ’s meaning then is, “God only knows what
year and day and hour the end of the world and the Judgment
shall be. And although God has caused Me, Christ,
as I am man, to know the same, as I am that one man who
is united to the WORD; yet as I am the Father’s ambassador
to men, He has not willed Me to make known that
day, but to keep it secret, and to stir them up continually
to prepare themselves for it.” There is a like mode of expression
in St. John 15:15.