Revelation chapter 12 presents the Red Dragon as having seven heads and ten horns, while Revelation 13 depicts the beast in mirror image as having ten horns and seven heads.  As such, they are reflections of the same subject.  The "crowned" heads of the Red Dragon refer to seven provincial rulers and the ten horns refer to ten Roman Emperors.  The beast, in reverse, has the "crowned" horns referring to seven Roman Emperors and the heads to ten provincial rulers.

The Red Dragon

From the testimony of the ancient historians Suetonius, Cassius, and Josephus, we learn that the early Romans counted Julius Caeser, not Augustus, as the first Roman emperor.  So by this reckoning, although ultimately there were scores of  Roman emperors, only ten ruled from the beginning of the Roman empire till the time of the Jewish revolt.  The Jewish revolt lasted seven years, though the sacrifices were halted and the temple destroyed after only three and a half years, in the year 70AD.  

Conversely, the seven heads in this instance represent the Herodian dynasty.  It may seem strange that the provincial Herodian dynasty - the red dragon - is identified as Satan (Revelation 12:9).  However, to put this in perspective, even the apostle Peter was identified as Satan at one point (Matthew 16:23).  The "red" designation reflects Herod's Idumean, that is Edomite, ancestry - wherein "edom" means "red."  In the table below, many reasons are given why these provincial rulers received so much attention in Revelation chapter twelve, but probably the most significant of those reasons belonged to king Herod's attempt to kill Jesus as an infant (Revelation 12:3-6):

The above arrangement is reflected also in Daniel's prophecies:

Daniel 7:
7 After this I saw in the night visions, and behold, a fourth beast, terrible and dreadful and exceedingly strong; and it had great iron teeth; it devoured and broke in pieces, and stamped the residue with its feet. It was different from all the beasts that were before it; and it had ten horns.
8 I considered the horns, and behold, there came up among them another horn, a little one, before which three of the first horns were plucked up by the roots; and behold, in this horn were eyes like the eyes of a man, and a mouth speaking great things...
23 "Thus he said: `As for the fourth beast, there shall be a fourth kingdom on earth, which shall be different from all the kingdoms, and it shall devour the whole earth, and trample it down, and break it to pieces.
24 As for the ten horns, out of this kingdom ten kings shall arise, and another shall arise after them; he shall be different from the former ones, and shall humble (abase) three kings.
25 He shall speak words against the Most High, and shall wear out the saints of the Most High, and shall think to change the times and the law; and they shall be given into his hand for a time, two times, and half a time.

The "three kings" who were "plucked and humbled" refer to Galba, Otho, and Vitellius.  The little horn may be Vespacian or his son Titus, but it more likely refers to the antichrist - Ananus son of Ananus.  As a leading figure in the Jewish revolt, Ananus endorsed defiance of Roman authority by his denigration of the three weak emperors.  As high-priest, he enthusiastically persecuted Christians and had James the apostle executed.

Finally, the prophetic statue of gold (Babylon), silver (Medo-Persia), bronze (Greece), and iron (Rome) is described by Daniel as having feet of iron mixed with clay.  This represents a "divided kingdom" (2:41) which is "partly strong and partly brittle where they will mix with one another in marriage but they will not hold together" (2:42-43).  Regarding mixed marriages, the Law commands "you shall not intermarry with them (non-Jews), for they would turn your sons from following me to serving other gods" (Deuteronomy 7:3-4, Ezra 9, Nehemiah 9:2).  Nevertheless the Roman provincial rulers, some Jewish and some not, had accumulated many of these mixed marriages which predictably diluted the faithfulness of both the rulers and their subjects.  This was particularly true in the house of Herod.  And ironically, emperor Nero himself succumbed and was persuaded to persecute Christians by his Jewish wife Poppea.

Eventually the intermingling of Jew and Roman would last no longer, and as Daniel prophesied, Rome and Jerusalem turned against each other (2:41-43).

The Beast

As noted earlier, the beast of Revelation chapters 13 and 17 mirrors the dragon wherein "ten kings" now refers to provincial kings of the Roman empire and "seven horns" this time refers to Roman emperors:

Revelation 17:9 This calls for a mind with wisdom: the seven heads are seven mountains (seven hills of Rome) on which the woman is seated;
10 they are also seven kings, five of whom have fallen, one is (Nero), the other has not yet come, and when he comes he must remain only a little while (Galba).
11 As for the beast that was and is not, it is an eighth but it belongs to the seven, and it goes to perdition.
12 And the ten horns that you saw are ten kings who have not yet received royal power, but they are to receive authority as kings for one hour, together with the beast.
13 These are of one mind and give over their power and authority to the beast;

The attention given to Nero above is found also in the thirteenth chapter: "I saw a beast rising out of the sea, with ten horns and seven heads...One of its heads seemed to have a mortal wound, but its mortal wound was healed, and the whole earth followed the beast with wonder" (Rev 13:1.3).  Nero commited suicide by stabbing himself in the throat while a Praetorian Guard assisted in finishing him off.  Nevertheless, he was proclaimed a god by the Senate and worshipped as such; there was widespread rumor that Nero would be resurrected to life.  Finally, Revelation 13:18 states "the number of the beast" as 666 which corresponds to the Greek-to-Hebrew numerology of the name "Caeser Nero."  And accordingly, in some ancient book of Revelation manuscripts, the number of the beast is stated instead as 616 - which is the numerology of "Caeser Nero" when rendered in Latin.