The accidentally ordered series of efficient causes is the kind of series that St. Thomas thinks can logically extend backwards into the past. There is nothing which limits the causal series, since each cause has its power “built-in.” We don’t have to come to a “first,” since each, so to speak, has it’s own “first-ness built-in.” Their power of causation is not immediately derived from the mover that moved them to have that power of causation. As a first example, let’s say A causes B. Then A ceases to exist. Now, B causes C. That would be an accidentally ordered series. The reason is that B’s causal power to produce C is not immediately derived from A; since, when A stopped existing, B still had the power to cause C. To use a more concrete example, a father begets a son. The father (barring biological technicality here) is the efficient cause of the son; the father causes the son to exist. But is the son’s own power to be an efficient cause of his own son (the grandchild) immediately dependent upon the father? Certainly not; Grandpa could be dead, but this does not prevent the son from bearing his own son (the grandchild). The power of the son to cause his own son is “built-in” to the son; it is not immediately borrowed from the father, even if, at one point it time, it was in fact gotten from him.
Any time we are dealing with an essentially ordered series of efficient causes, there must be a “first,” in the sense that there must be an efficient cause that does not derive it’s causal power from something anterior. Otherwise, every cause in the series derives from something else, and there’s nothing to derive from. Hence, they derive nothing, and they actually would not be causing anything. But they *are *causing, which is what incites the question to being with, and so they must derive their causal power from somewhere, since it does not come form them. This means that, in the series, there must actually be an efficient cause which does not derive its causal power.
Imagine I balance Aristotle’s Metaphysics on top of Wheelock’s Latin, on top of Dostoevsky’s Crime and Punishment. Wheelock’s Latin does not have the “built-in” power to hold up Aristotle’s Metaphysics in the same way a son has the “built-in” power to beget his own son. If we take out Crime and Punishment (without replacing it with an equivalent), then both Aristotle’s Metaphysics and Wheelock’s Latin will fall. It’s not like taking Grandpa out of the picture (sorry Grandpa); causally speaking, there is a certain sense in which we can do without Grandpa, whereas we cannot do without Crime and Punishment. That “certain sense” has to do with whether the causal power is derived or “built-in.” Those are the two options. Does the efficient cause derive its power from an immediately anterior efficient cause in the series? If no, then we’re dealing with an accidentally ordered series of efficient causality, and the power is “built-in” to the efficient cause such we can do without Grandpa. If yes, then we’re dealing with an essentially ordered series of efficient causality, such that the power is not “built-in” to the cause but derived from an immediately anterior one, such that we cannot do without Crime and Punishment.
And in fact, that still begs the question, because Crime and Punishment would still be held up by my hands, which is held up by the rest of my body, the floor, and finally the earth. When we start dealing with the earth, we start dealing with things like inertia and other forces, so we have an accidentally ordered series of efficient causes. But the point is that, when there is an essentially ordered series of efficient causes, there must be a first cause that has its power “built-in.” St. Thomas argues that creation is itself an essentially ordered series. Therefore, there must be a first in the sense of “underived,” which of course applies even if the universe is infinitely old. No matter hold old the universe is, that does not make it the kind of thing that has underived actuality. Otherwise it would be pure act, and it would be God. But that is clearly not the case, because there would be no observable, change, contingency, or distinction whatsoever.