In Aquinas's first proof of God's existence, what is the difference between accidentally and essentially ordered causal series?

Here's S.Th., I, q.46, a.2, ad 7, for reference:

In efficient causes it is impossible to proceed to infinity "per se"—thus, there cannot be an infinite number of causes that are "per se" required for a certain effect; for instance, that a stone be moved by a stick, the stick by the hand, and so on to infinity. But it is not impossible to proceed to infinity "accidentally" as regards efficient causes; for instance, if all the causes thus infinitely multiplied should have the order of only one cause, their multiplication being accidental, as an artificer acts by means of many hammers accidentally, because one after the other may be broken. It is accidental, therefore, that one particular hammer acts after the action of another; and likewise it is accidental to this particular man as generator to be generated by another man; for he generates as a man, and not as the son of another man. For all men generating hold one grade in efficient causes—viz. the grade of a particular generator. Hence it is not impossible for a man to be generated by man to infinity; but such a thing would be impossible if the generation of this man depended upon this man, and on an elementary body, and on the sun, and so on to infinity.

St. Thomas distinguishes between causal series ordered per accidens ("accidentally") and causal series ordered per se ("essentially"). Edward Feser's Aquinas: A Beginner's Guide (ch. 3 "Natural Theology," § "The First Way") describes the difference:

This brings us to an important distinction Aquinas and other medieval thinkers made between two kinds of series of efficient causes. On the one hand there are causal series ordered per accidens or “accidentally,” in the sense that the causal activity of any particular member of the series is not essentially dependent on that of any prior member of the series. Take, for example, the series consisting of Abraham begetting Isaac, Isaac begetting Jacob, and Jacob begetting Joseph. Once he has himself been begotten by Abraham (and then grows to maturity, of course), Isaac is fully capable of begetting Jacob on his own, even if Abraham dies in the meantime. It is true that he would not have existed had Abraham not begotten him, but the point is that once Isaac exists he has the power to beget a son all by himself, and Abraham’s continued existence or non-existence is irrelevant to his exercise of that power. The same is true of Jacob with respect to both Abraham and Isaac, and of Joseph with respect to Abraham, Isaac, and Jacob. Given that we are considering them as a series of begetters specifically, each member is independent of the others as far as its causal powers are concerned. Contrast this with a causal series ordered per se or “essentially.” Aquinas’s example from the First Way of the staff which is moved by the hand is a standard illustration, and we can add to the example by supposing that the staff is being used to move a stone, which is itself moving a fallen leaf. Here the motion of the leaf depends essentially on the motion of the stone, which in turn depends essentially on the motion of the staff, which itself depends essentially in turn on the motion of the hand. For if any member higher up in the series ceases its causal activity, the activity of the lower members will necessarily cease as well. For instance, if the staff was to slip away from the stone, the stone, and thus the leaf too, will stop moving; and of course, if the hand stops moving, the whole series, staff included, will automatically stop. In this case the causal power of the lower members derives entirely from that of the first member, the hand. In fact, strictly speaking it is not the stone which is moving the leaf and the staff which is moving the stone, but rather the hand which is moving everything else, with the stone being used by it as an instrument to move the leaf and the staff being used as an instrument to move both stone and leaf.

Causal series ordered per accidens are linear in character and extend through time, as in the begetting example, in which Abraham’s begetting Isaac occurs well before Isaac’s begetting Jacob, and Isaac’s begetting Jacob occurs well before Jacob’s begetting Joseph. Causal series ordered per se are paradigmatically hierarchical with their members acting simultaneously, as in the staff example where the movement of the leaf occurs precisely when the movement of the stone occurs, which is precisely when the movement of the staff occurs, which is precisely when the movement of the hand occurs. Now it is in Aquinas’s view at least theoretically possible for a causal series ordered per accidens to regress to infinity, and thus have no beginning point (ST I.46.2). (This is why Aquinas thinks it is not possible to prove via purely philosophical arguments that the world must have had a beginning in time.) For since each member of such a series has its causal power independently of the earlier members, there is no need to trace any particular member’s action back to the activity of a first member; for instance, when Jacob begets Joseph, it is precisely Jacob who begets him, and not Abraham who begets him by using Isaac and Jacob as instruments. By contrast, “in efficient causes it is impossible to proceed to infinity per se – thus, there cannot be an infinite number of causes that are per se required for a certain effect; for instance, that a stone be moved by a stick, the stick by the hand, and so on to infinity” (ST I.46.2). For “that which moves as an instrumental cause cannot move unless there be a principal moving cause” (SCG I.13.15). That is to say, since the lower members of a causal series ordered per se have no causal power on their own but derive it entirely from a first cause, which (as it were) uses them as instruments, there is no sense to be made of such a series having no first member. If a first member who is the source of the causal power of the others did not exist, the series as a whole simply would not exist, as the movement of the leaf, stone, and staff cannot occur in the absence of the hand.

What Aquinas is saying, then, is that it is in the very nature of causal series ordered per se to have a first member, precisely because everything else in the series only counts as a member in the first place relative to the actions of a first cause. To suggest that such a series might regress infinitely, without a first member, is therefore simply unintelligible. The leaf is “moved” by the stone only in a loose sense; strictly speaking, the leaf, stone, and staff are all really being moved by the hand. Thus to suggest that this series of purely instrumental causes might regress to infinity is incoherent, for they would not in that case be the instruments of anything at all (CT I.3). As A. D. Sertillanges put it, you might as well say “that a brush can paint by itself, provided it has a very long handle” (quoted by Garrigou-Lagrange in God: His Existence and His Nature).

Given their essentially instrumental character, all causes in such a series other than the first cause are referred to by Aquinas as “second causes” (“second” not in the sense of coming after the first but before the third member of the series, but rather in the sense having their causal power only in a secondary or derivative way). It is worth emphasizing that it is precisely this instrumental nature of second causes, the dependence of whatever causal power they have on the causal activity of the first cause, that is the key to the notion of a causal series per se. That the members of such a series exist simultaneously, and that the series does not regress to infinity, are of secondary importance. As Patterson Brown and John Wippel point out, even if a series of causes ordered per se could somehow be said to regress to infinity, it would remain the case, given that they are merely instrumental causes, that there must then be something outside the entire infinite series that imparts to them their causal power.

Whether or not the series of causes per accidens regresses infinitely into the past, then – and again, while Aquinas believed that it did not, he didn’t think this could be proven through philosophical arguments – a causal series per se existing here and now, and at any moment we are considering the matter, must necessarily trace back to a first member. But strictly speaking, even the hand in Aquinas’s example doesn’t count as a first mover – the example is intended merely as a first approximation to the notion of a first mover – because it is itself being moved insofar as its activity depends on the motion of the arm, the flexing of certain muscles, and so forth. To understand the way in which such a series regresses and how it does and must terminate, it is crucial to remember that for Aquinas, motion or change is just the reduction of potency to act. So when we talk about one thing being moved by another, which is moved by another, and so on, in a causal series ordered per se, this is shorthand for saying that a certain potency is reduced to act by something whose potency is itself reduced to act by something whose potency is itself reduced to act by … and so forth. (Or, to soften the technical terminology slightly, a certain potentiality is actualized by something whose potentiality is itself actualized by something whose potentiality is itself actualized by … and so on.) As should be evident, such a series can only possibly terminate in something which is not reduced to act or actualized by anything else, but which just is in act or actual, and thus “unmoving.” The potential of the hand for movement is actualized here and now by the flexing of the muscles of the hand, the potential of the muscles to flex is actualized here and now by the firing of certain motor neurons, the potential of the motor neurons to fire is actualized here and now by the firing of certain other neurons, and so forth. Eventually this regress must terminate in something which here and now actualizes potentialities without itself being actualized, an unmoved mover.