The view given below mainly accords with what others call the "Allegorical" view, but the author states his book as being "The Resumptive Interpretation", described like this:
"With such an interpretation the book [of Revelation] applies to all
ages of the church, and all the book applies throughout those ages.
Revelation appears as a vast, unfolding, an unveiling, discovering the
great but unseen principles, the massive but invisible powers, that
lie behind the outward appearance of history, and the visible
occurrence of events.
Both principles and powers being invisible, that are depicted with
graphic imagery, stunning the imagination, in which two opposing
forces, in conflict the one with the other, range over the heavens and
the earth.
This titanic conflict takes on a variety of aspects. Hence the book
repeats the period from the ascension till the judgment over and over
again, consistently returning to the same beginning, and, from yet
another perspective, arriving at the same conclusion. This happens
seven times over.
Hence the term, 'Resumptive'. The period is resumed but under a
different standpoint. The entire age of the church appears in the
spiritual - but intangible - verities which govern it, so as to
display seven times over, from one prospect or another, seven parallel
phases of conflict between implacably antagonistic principles.
Thus, allowing for prologue, intermissions, and epilogue, the
sevenfold parallel sections of the Book of the Revelation, each
concluding with the last judgment, cover the same vast era of time,
yet in ever ascending and stupendous openings of distinctive aspects
of the same struggle." The Revelation of Jesus Christ, John
Metcalfe, pp.15-16
http://www.johnmetcalfepublishingtrust.co.uk/contact_us.htm
Given that the author has previously expounded three such 'openings', this bit might be difficult for a new-comer to pick up. Further, it is then necessary to delve into the prophetic bits of Daniel to substantiate what these seven heads collectively represent. Here are some relevant extracts:
"Chapter 13 reveals the two global concepts - depicted as powers or
authorities - by which the adversary deceives the whole world, and
challenges Christ lawfully to recover the throne and dominion which he
has usurped by spiritual wickedness.
The powers, authorities, or means by which this wickedness is
established and sustained are depicted as two beasts. This is the
allegory. Consider the first [as in the Rev. 13:1-3 quote in the
question, above]
...as always, the root lies in the Old Testament. And these precedents
are to be found in the visions and prophecies of Daniel... If these
beasts seen in vision by Daniel had not the very features combined in
the first beast of Revelation chapter 13, what else do these
descriptions convey? Even so to 'a mouth speaking great things',
Daniel 7:8, agreeing with 'and there was given unto him a mouth
speaking great things', Revelation 13:5. But there is this difference.
Daniel saw a series of figurative beasts in succession: first a
lion; then a bear; next a leopard; finally a dreadful beast having
iron teeth.
Not so the vision of the beast in Revelation 13. There is no
succession here. There is an amalgamation. In Revelation 13 the
beast is a composite... Daniel's series of beasts combine to give form
to the beast of Revelation... Now these visions are some six hundred
and fifty years apart, yet in so vast an interval everything Daniel
had prophesied had come to pass... This one beast evidently
incorporated the separate and distinct four beasts seen by Daniel,
since features peculiar to each were common to that one beast seen by
John.
But, whether Daniel or John, all arose out of the sea, that is, the
waters. "And he said unto me, The waters which thou sawest, where the
whore sitteth, are peoples, and multitudes, and nations, and tongues."
Both Daniel and John saw strange symbolic forms arise out of the
waters. The sea being in a state of turmoil, it signifies the nations,
allegorically viewed as striving without effect the one against the
other... a symbol of chaotic heaving and movement among the
multitudes, a state common to the greater part of history. But not to
all history. Ever and anon, out of such turbulent chaos, form
arises... Although Daniel saw but one beast rise after the other, John
discerns all these beasts as being various successive manifestations
of but one.
...Of these 'heads', Daniel had envisaged four... John perceives a
deeper principle; the heads might have come one after the other, they
might have appeared diverse, but behind then all lay one single
concept; the beast."[Ibid. pp.304-308]
All these snippets can do is whet your appetite to discover more. And there are about 600 more pages of exposition, starting at the beginning of Revelation, through to the end. Don't expect a short little answer to this particular beast out of the sea.
EDIT re. comment: The other verses about this beast come in Revelation 17:7-14, as quoted by the o.p. However, chapters 17 to 19 form the Sixth Opening (or, 'sixth back to the beginning' section). My answer was confined to the Fourth Opening section. To jump back and forward with verses as if on a yo-yo string is not the way to work out the immense complexity involved. A picture has to be gradually built up, like layers of washes of paint on a canvass. When that it done, eventually the full picture might be seen. But on this site, only a snippet can ever be presented to such questions. Hopefully, the 600+ page book will be read, which even the commentator will be pleased to see describes in much more detail what was supposed about the woman who sits on that beast. This question was not asking about her, though.
