Aquinas's Second Way to prove the existence of God (from newadvent.org):
The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.
A common way to interpret this passage is to claim that Aquinas is talking about per se causal series in this argument, not per accidens causal series. A per se causal series is one in which the later causes depend on the prior causes essentially; that is, the later causes stop causing when the prior causes stop causing. The standard example is an arm pushing a stick pushing a rock. When the arm stops pushing, the stick stops pushing the rock.
This is contrasted with the per accidens causal series, in which the later causes can continue causing even when the prior causes cease. The example here is Abraham begetting Isaac. Isaac himself can beget even if Abraham dies. He doesn't cease being causal when his own cause ceases.
The reason Aquinas is generally interpreted as speaking of per se causal series in this argument is that he also famously held that per accidens efficient causal series could in fact be infinite. He thought the world in principle could be eternal, although he thought it in fact had a beginning because of Scripture.
But I'm not sure we have to interpret Aquinas this way. Consider a series of dominos knocking themselves down. We can ask of any particular domino: What caused it to fall? And we answer: The domino prior to it in the series. And we can ask of that domino what caused it to fall, and again we get the answer: The domino prior to it in the series.
Suppose the row of dominos only consists of one domino. What is the answer to the question What caused the domino to fall? Since there is only one domino, the answer is: Something external to the row of dominos itself. Suppose the row of dominos consists of two dominos. The second domino falls as a consequence of the first domino falling, and with that first domino, we are left in exactly the same place as we were with the row consisting of a single domino. Something outside of the row of dominos itself must have caused it to fall. The only thing the additional domino did in the row of two dominos vs the single domino is simply to displace the question to another domino. It did nothing to answer the essential question.
Now consider a row of dominos of arbitrary length. What do all those extra dominos do as far as answering the essential causal question? They do nothing other than displace the question from one domino to the next. Even were the row of dominos infinite, all that would do is displace the question an infinite distance. It wouldn't get us one inch closer to answering the question of what caused the dominos to fall than we were with the one or two domino series. So the fact that a per accidens causal series might be infinite in length is irrelevant to the causal question asked in Aquinas's Second Way.
Suppose God created a world with an infinite series of dominos in it, that is in the process of falling. In the act of creation, he would have to pick one or more dominos somewhere in the series to get the falling process rolling. Call that domino D1. God creates D1 in the state of falling. Now God could create all dominos prior to D1 in the fallen state, and all dominos after D1 in the standing state. Were he to create all the dominos in a standing state, then the row of dominos would remain in the standing state eternally, whatever its length. Looking at the series after the fact of creation, we see an infinite series of fallen dominos disappearing in the distance in one direction, and an infinite series of standing dominos in the other direction, with the dominos in front of us in the process of falling. This might give us the illusion that the dominos were falling forever and therefore had no need of an exterior cause to get the process of falling going. But that is a distraction. Whether they were falling forever or not, something external to the row of dominos itself was the ultimate origin of the state of falling.
I think it is important to note that the "first efficient cause" in the series can't be just another efficient cause. If it were, then it would need the same prior causal explanation that all the other efficient causes did. The "first efficient cause" must be something over and beyond the causes in the series in question. The point of Aquinas's proofs, after all, is to show that nature points beyond itself to a supernatural origin. In this case, the series of efficient causes in nature points beyond itself to a "first efficient cause" that jumpstarts the efficient causal series that we experience in nature.
