Just to reinforce the point others above have made about this apologist piece. I went through the lot, trying to find on what basis the author supports the dates of 539 BC and 537 BC. The only thing I could find, was the following, on about page 12:
Of course, there is abundant secular evidence to show that 539 BCE is the date of Babylon’s defeat, and most importantly, such evidence is accepted by us because it does not contradict the scriptures.
So the whole piece rests on that. Accept the secular 539 BC datum and reject the secular 587 BC datum. No explanation as to how one can be accepted and the other rejected.
On another thread, I reported that I went back through the old literature for where 537 BC came from. The history is lengthy. I will try to very briefly summarise.
Russell assumed the dates of 536 BC and 606 BC. Back then, historians had already known for centuries (from Ptolemy's canon) that the date Cyrus conquered Babylon in around 538 or 539 BC (and that Jerusalem fell in about 587 BC), but the date of 536 BC had a special numerical allure for the Millerites. Millerites assumed Ptolomy's calendar was out by 2 years, not understanding that that was impossible, because the timing of the eclipses in Ptolomy's canon could not be out that far. Russell just copied the 536 date as fact, from around the 1890's onwards, without checking.