I recently commented on a thread in relation to the 607 BC - 1914 AD calculation, listing problems with the logic and mathematics. Afterwards, I wondered what made CT Russell choose 606 BC in the first place. Here is my research so far.
I had read somewhere a suggestion that 606 BC was a number CT Russell got from Nathan Balbour who in turn got it from William Miller of the Millerites. I think I also read somewhere that there was also a mixup along the way because Miller used that date as when Nebuchadnezzar II ascended the throne, not when Jerusalem fell. At the moment, I can't confirm any of that. Perhaps someone else can.
CT Russell's own answer to the question appears in the second book of his "Studies in the Scriptures" series. That book, titled "The Time is at hand" (first published 1889) contains the following passage in Study IV:
The Bible evidence is clear and strong that the "Times of the Gentiles" is a period of 2520 years, from the year B.C. 606 to and including A.D. 1914. This lease of universal dominion to Gentile governments, as we have already seen, began with Nebuchadnezzar--not when his reign began, but when the typical kingdom of the Lord passed away, and the dominion of the whole world was left in the hands of the Gentiles. The date for the beginning of the Gentile Times is, therefore, definitely marked as at the time of the removal of the crown of God's typical kingdom, from Zedekiah, their last king.
According to the words of the prophet (Ezek. 21:25-27), the crown was taken from Zedekiah; and Jerusalem was besieged by Nebuchadnezzar's army and laid in ruins, and so remained for seventy years--until the restoration in the first year of Cyrus. (2 Chron. 36:21-23) Though Jerusalem was then rebuilt, and the captives returned, Israel has never had another king from that to the present day. Though restored to their land and to personal liberty by Cyrus, they, as a nation, were subject successively to the Persians, Grecians and Romans. Under the yoke of the latter they were living when our Lord's first advent occurred, Pilate and Herod being deputies of Caesar.
With these facts before us, we readily find the date for the beginning of the Gentile Times of dominion; for the first year of the reign of Cyrus is a very clearly fixed date--both secular and religious histories with marked unanimity agreeing with Ptolemy's Canon, which places it B.C. 536. And if B.C. 536 was the year in which the seventy years of Jerusalem's desolation ended and the restoration of the Jews began, it follows that their kingdom was overthrown just seventy years before B.C. 536, i.e., 536 plus 70, or B.C. 606. This gives us the date of the beginning of the Times of the Gentiles--B.C. 606.
There are two obvious issues with Russell's purported use of Ptolemy's Canon.
1. Ptolomy's Canon does not have that date
Russell asserts Ptolemy's Canon shows the first year of Cyrus's reign as 536 BC. It doesn't. It shows the first year to be 538 BC (or Egyptian year NE 210). Note that the way Ptolemy's canon works, Cyrus becoming ruler of Babylon in late 539 BC (12 or 13 Oct 539 BC) is not an issue. There is only one named King per year, and if a King commences a reign late in the year, the year is still in the name of the previous King.
Russell did not copy this date from Miller. Miller, writing 50 years earlier, seems to have chosen the correct dates, probably from Ptolomy's Canon. On one of Miller's pictorials available on Wikipedia, the date for the fall of Babylon is shown as 538 BC.
I suspect that Russell got the date of 536 BC (probably indirectly) from Bishop Ushher's "The Annals of the World", written 1658 (yes, it really that old). Bushop Ushher did write that the end of the 70 years was 536 BC (but also wrote that Cyrus conquered Babylon in 538 BC).
2. It is obvious from Ptolomy's canon that 606 BC can't be correct
If Russell had Ptolomy's canon in front of him when he wrote the above passage, he should have noticed Nebuchadnezzar II on the list, six kings, and just 66 years earlier. It records Nebuchadnezzar II commencing his reign on 604 BC.
2 Kings 25:8-9 records the date of the destruction of Jerusalem as follows:
In the 5th month, on the seventh day of the month, that is, in the 19th year of King Nebu Echad Enezzar the king of Babylon, Nebeuzareadane the chief of the guard, the servant of the king of Babylon, came to Jerusalem. He burned down the house of Jehovah, the king’s house, and all the houses of Jerusalem; he also burned down the house of every prominent man.
If Russell actually knew his bible and had Ptolemy's Canon in front of him, he would have simply added the 19 years and five months to the date Nebuchadnezzar became king, according to Ptolemy's Canon, and landed on around the generally accepted date for the fall of Jerusalem.
Some additional notes:
I understand that some KJV bibles used to include Bishop Ushher's chronology, up to around the end of the 19th century. I presume that by the 20th century, a chronology showing the universe was created in 4004 BC lacked credibility. Perhaps Russel got his 536 BC date from one of those bibles.
The fact that Russell refers to Ezekiel 21:25-27 in the second para of the above quote, rather than something out of Jeremiah or 2 Kings to support his argument, makes me wonder if CT Russell was much of a bible scholar, at all.
66 years being less than 70 years is not necessarily a problem with the scriptures. If you read Jerimiah carefully, it is reasonable to interpret the 70 years beginning while Nebuchadnezzar's father while still on the throne.
You can purchase or read free copies of "The Time is at Hand" and the other literary treasures of the Studies in Scriptures series, replete with pyramid sketches, at the website of the Bible Students (a.k.a. "the evil slave").
Most non-JW Christians who take the bible literally calculate the fall of Jerusalem to be either 586 BC or 587 BC, near to or the same as secular and archeological sources. An example of biblical study leading to 586 BC (and what proper scholarly research looks like) is at:
In "The Annals of the World" (1658), Bishop Ushher recorded the reign of Nebuhadnezzer as commencing 605 BC and the fall of Jerusalem as having occured in 588 BC.
William Miller must have used Ptolomy's Canon. He chose his start date (the decree to rebuild Jerusalem) as 457 BC, which is the start of the reign of Artaxerxes I as 464 BC plus 7 years in accordance with Ezra 7:7-8. If he had used Bishop Ushher's dates, he would have been 3 years out, or more, depending on how he read it.
To me, Miller's postulations look far more convincing than CT Russell's. Daniel 8:14 reads much more like an end time prophecy, and his calculation to get to 1843 AD doesn't rely on 2 types of year being combined. However, Miller also appears to have forgotten that there is no zero AD.