What's the difference between Galileo's "impeto" and "momento"?

Question

In Galileo's Two New Sciences, he describes an experiment demonstrating pendulum motion and how the pendulum will rise to the same height from where it started its fall. This discussion can be found just after Fig. 45 in this source. He describes the pendulum as acquiring an impeto, then later switches to mometo, then returns back to using impeto. For example:

...from this we may rightly infer that the ball in its descent through the arc CB acquired a momentum [impeto] on reaching B, which was just sufficient to carry it through a similar arc BD to the same height. ... From this we can see what can be done by the same momentum [impeto] which previously starting at the same point B carried the same body through the arc BD to the horizontal CD.

...for since the two arcs CB and DB are equal and similarly placed, the momentum [momento] acquired by the fall through the arc CB is the same as that gained by fall through the arc DB; but the momentum [momento] acquired at B, owing to fall through CB, is able to lift the same body [mobile] through the arc BD; therefore, the momentum [momento] acquired in the fall BD is equal to that which lifts the same body through the same arc from B to D; so, in general, every momentum [momento] acquired by fall through an arc is equal to that which can lift the same body through the same arc. But all these momenta [momenti] which cause a rise through the arcs BD, BG, and BI are equal, since they are produced by the same momentum [momento], gained by fall through CB, as experiment shows. Therefore all the momenta [momenti] gained by fall through the arcs DB, GB, IB are equal.

...although the above experiment shows us that the descent of the moving body through the arc CB confers upon it momentum [momento] just sufficient to carry it to the same height through any of the arcs BD, BG, BI, we are not able, by similar means, to show that the event would be identical in the case of a perfectly round ball descending along planes whose inclinations are respectively the same as the chords of these arcs.

...In striking these planes some of its momentum [impeto] will be lost and it will not be able to rise to the height of the line CD; but this obstacle, which interferes with the experiment, once removed, it is clear that the momentum [impeto] (which gains in strength with descent) will be able to carry the body to the same height.

Most translations I've seen (like the one above) translate both impeto and momento as being "momentum," but during his time, did Galileo believe impeto to be different from momento? If so, in what way were they different?

I understand impeto to be impetus (the driving "force" of an object in motion), but I'm having difficulty finding support for momento since Galileo never defines it in his text. I've found that several articles that briefly mention momento, but they draw no comparison to his use of impetus. For example, in the Stanford Encyclopedia of Philosophy for Galileo, it states:

A while later, in his 1600 manuscript version of Le Meccaniche (On Mechanics), Galileo introduced the concept of momento, a quasi force that applies to a body at a moment, and which is somehow proportional to weight or specific gravity (Galluzzi 1979). Still, he had no good way to measure or compare specific gravities of bodies of different kinds, and his notebooks during this early seventeenth-century period reflect his trying again and again to find a way to bring all matter under a single proportional measuring scale. He tried to study acceleration along an inclined plane and to find a way to think of what changes acceleration brings to momento.

From the above passage, it's interesting to note that most of the articles I've come across always cite Italian historian Paolo Galluzzi and his 1979 book momento, which might shine some light on this question; however, it seems the book only exists in Italian (I'm not fluent enough for it to be practical).

Speaking of Le Meccanich, I came across another related post about momentum where user Maruo ALLEGRANZA cites passages from Le Meccanich discussing momento; however, the specific section (definitions [diffinizioni]) that he quotes is conveniently missing but the rest of the text is functional. The missing section makes it difficult to understand the context of the quoted passage.

I also found that Galileo discusses momento in his Discorse of Floating Objects [Discorso intorno alle cose che stanno in su l'acqua, o che in quella si muovono] where momento is translated as "moment", as seen in Definition V and Axioms I-III. That led me to the history section of moment which suggests that it has origins stemming from momento:

"That said, why was the word momentum chosen for the translation? One clue, according to Treccani, is that momento in Medieval Italy, the place the early translators lived, in a transferred sense meant both a "moment of time" and a "moment of weight" (a small amount of weight that turns the scale).

In 1554, Francesco Maurolico clarifies the Latin term momentum in the work Prologi sive sermones. Here is a Latin to English translation as given by Marshall Clagett:

[...] equal weights at unequal distances do not weigh equally, but unequal weights [at these unequal distances may] weigh equally. For a weight suspended at a greater distance is heavier, as is obvious in a balance. Therefore, there exists a certain third kind of power or third difference of magnitude—one that differs from both body and weight—and this they call moment.[c] Therefore, a body acquires weight from both quantity [i.e., size] and quality [i.e., material], but a weight receives its moment from the distance at which it is suspended. Therefore, when distances are reciprocally proportional to weights, the moments [of the weights] are equal, as Archimedes demonstrated in The Book on Equal Moments.[d] Therefore, weights or [rather] moments like other continuous quantities, are joined at some common terminus, that is, at something common to both of them like the center of weight, or at a point of equilibrium. Now the center of gravity in any weight is that point which, no matter how often or whenever the body is suspended, always inclines perpendicularly toward the universal center.

In addition to body, weight, and moment, there is a certain fourth power, which can be called impetus or force.[e] Aristotle investigates it in On Mechanical Questions, and it is completely different from [the] three aforesaid [powers or magnitudes]. [...]" (emphasis added)

Again, I only find brief mentions that impetus and moment are different, but no elaboration.

This whole thing has turned into one nasty rabbit hole full of nested sources, deadends, or missing/unavailable sources, and I'm ultimately left with a very nebulous understanding of momento and what Galileo intended. Considering what I've found in the above, it just leaves me confused how "moment" relates to Galileo's pendulum experiment. If I assume he means "moment" as described in the moment wiki article (which is plausible since the pendulum is rotating about a fixed point), then it kind of makes sense, especially when he discusses levers in other sections of the book, but later on, it doesn't make sense at all. For example, in Two New Sciences, Galileo says,

"Now seeing how great is the resistance which the air offers to the slight momentum [momento] of the bladder and how small that which it offers to the large weight [peso] of the lead, I am convinced that, if the medium were entirely removed, the advantage received by the bladder would be so great and that coming to the lead so small that their speeds would be equalized."

and

I (i.e., Simplicio) am one of those who accept the proposition, and believe that a falling body acquires force [vires] in its descent, its velocity increasing in proportion to the space, and that the momentum [momento] of the falling body is doubled when it falls from a doubled height

and

The speed reaches a maximum along a vertical direction, and for other directions diminishes as the plane diverges from the vertical. Therefore the impetus, ability, energy, [l’impeto, il talento, l’energia] or, one might say, the momentum [il momento] of descent of the moving body is diminished by the plane upon which it is supported and along which it rolls.

in which neither don't make sense in the context of "moment" since these are descriptions of linear motions. I'm at a loss...

Answer 1

Momento is an object’s tendency to move, much like a net force acting on an object that results in motion. For example, lighter objects move slower in a medium than heavier objects in the same medium. In this case, Galileo states the lighter object has a lesser momento on account of its lesser weight and motion in the medium. He also demonstrates that a body hanging over an inclined plane exhibits its entire momento and can be used to compare the partial momento of another body suspended on an inclined plane by coupling the two bodies together, and the equilibrium found between these weights and their respective lengths of the incline plane form what we now call the law of inclined planes.

Galileo originally developed momento in his Le Meccaniche as a static torque in his study of levers, to which he equated it with static linear forces. More specifically, by finding equilibrium for a lever that’s bent at the fulcrum, Galileo equated the static torque produced from a weight acting on the bent end of the lever with the static linear force acting on the same weight held in equilibrium on an inclined plane. He argued that the tangential forces on the bent lever in this example were substantially the same as the linear forces on the plane, but to Galileo, he called them both momento. Later on in Galileo’s writings, he uses momento for dynamics while referring to linear forces and torques without directly comparing each other, but the nexus still remains: in Two New Sciences, both his discussion of levers on the 2nd day and his linear acceleration of the 3rd day are both described using momento.

Galileo probably knew that momento was insufficient and ill-defined, and that it made leaping assumptions between statics and dynamics, thus prompting him (or more likely, his pupil Viviani) to insert in the second edition of the Two New Sciences (i.e., the National Edition) a section that gropes for alternate terms to describe incline motion, hence, “[l’impeto, il talent, l’engeria] or, one might say, [il momento] of descent of the moving body is diminished by the plane upon which it is supported and along which it rolls.”

Impeto is a combination of kinetic energy and the modern conception of momentum. Its effects can seen through impacts and is affected by the medium that is passes through. For example, Galileo discusses that the impeto of cannonball shot at point blank range is more destructive than when dropped from any height; and that the impeto is lost when a cannonball is shot into water and subsequently hits the ground softly. Impeto is capable of lifting objects to a height: cannonballs are shot upward with an impeto, and pendulums may be given impeto through successive puffs of air to start its swing. It is most similar to “impulse” in which an object is given impeto to change its velocity. He also states that objects moving from the same height down various inclined planes will each reach the bottom with equal impeto.

Source of Confusion: it's worth noting that the translation of momento and impeto as "momentum" was in an effort to make it mesh with modern physics terminology. In the translation I used, translators Henry Crew and Antonio Favaro note that:

Much of the value of any historical document lies in the language employed, and this is doubly true when one attempts to trace the rise and growth of any set of concepts such as those employed in modern physics. We have therefore made this translation as literal as is consistent with clearness and modernity. In cases where there is any important deviation from this rule, and in the case of many technical terms where there is no deviation from it, we have given the original Italian or Latin phrase in italics enclosed in square brackets. The intention here is to illustrate the great variety of terms employed by the early physicists to describe a single definite idea, and conversely, to illustrate the numerous senses in which, then as now, a single word is used. For the few explanatory English words which are placed in square brackets without italics, the translators alone are responsible.

However, in my opinion, drawing this equivalence made it incredibly misleading for modern readers for parsing Galileo's intentions since momento as a quasi-force in no way resembles that of a quantity of motion (i.e., "momentum"): Galileo states an object moving on a horizontal surface exhibits no momento, thus it doesn't make sense to call this momento as "momentum."

Sources (in no particular order):

"Galileo's Machines", Peter Machamer, The Cambridge Companion to Galileo

"The Use and Abuse of Mathematical Entities", Rivka Feldhay, The Cambridge Companion to Galileo

"Galileo's Perinertial Framework", Wallace Hooper, The Cambridge Companion to Galileo

"On Galileo's Writings on Mechanics: An Attempt at a Semantic Analysis of Viviani's Scholium", Halbwachs and Torunczyk (I found this paper to be the most helpful) https://www.jstor.org/stable/20116111

On Motion and On Mechanics, Comprising De Motu (ca. 1590) and Le Meccaniche (ca. 1600), Drabkin and Drake.

Concepts of Force: A Study in the Foundations of Dynamics, Jammer

Galileo, Courtier: The Practice of Science in the Culture of Absolutism, Biagioli

"Galileo", Stanford Encyclopedia of Philosophy

Answer 2

'Momentum' (Latin) or 'momento' (Italian) is always the momentum of X, where X is some quantity in its own right. Or so I would propose, at least for TNS. And when, in discussing motion, he just says 'momento,' I would suggest he means 'momentum of speed.'

The moment of a weight (see the Second Day) is its weight qua effective: e.g., two unequal weights at inversely proportional distances from the fulcrum have equal momenta of weight, but not equal weights. He talks about the moment that one position along a lever has, compared to another position: i.e., the effectiveness that a given weight would have at that position as compared to other positions.

So what does he mean by 'momenta of speed' in the treatment of accelerated motion? In one place he says "gradum seu momentum", as if to suggest they are equivalent, so that "degree of speed" = "momentum of speed." But if you look closely at the argument of his Proposition I on accelerated motion, he first argues that the aggregates of the parallel lines are equal, then that the degrees of speed are represented by those parallels, and only then does he bring in the term 'momentum,' concluding that the total momenta of speed in the accelerated motion are equal to the total momenta in the uniform motion. So here it looks like 'momentum of speed' is indeed the instantaneous value or degree of speed a body has at an instant, but also has the connotation of something like 'power to move a body a certain distance in a given time.' Otherwise it isn't clear what it adds to the line of argument to shift from 'degrees of speed' to 'momenta of speed.' Momentum of speed is the speed at an instant, qua effective at moving the body a given amount in a given time.

The common notion in the term, then, seems to be something causal, like this: the momentum of X is the instantaneous value of quantity X that a body has, insofar as that quantity is effective at bringing about some effect. The momentum of speed a body has at an instant is the value of its speed at that instant, considered as a power for the body to accomplish a certain motion as a result of that value. So you can see there is something quasi-inertial in the idea.

As for 'momentum' vs 'impetus', impetus is more like our modern momentum: it's measured by the impact a body is able to exert when it hits something. So the size (weight, mass, whatever) of the body matters for 'impeto', but not for 'momento' (where 'momento' is short for 'momentum of speed').

Here's the point that I think will help you: When Galileo is talking about one body only, he will use the terms 'momento' and 'impeto' interchangeably, since presumably the body isn't changing mass/weight, and he assumes (incorrectly) that the impact is proportional to the velocity.

Added note: I don't think it is correct to interpret him as saying that a body moving on a horizontal plane has no momentum of any kind. On my reading what he means is that it has no momentum of weight/heaviness: it still has a momentum of speed. The context governs which quantity the momentum is 'of': here he is comparing it to bodies going up or down inclined planes, so heaviness is the intrinsic causal quantity, whose effective value at an instant is the momentum.

I agree with Andrew R that Crew and De Salvo confuse things with their translation, but I think there is a reason they find it natural to translate momentum as momentum, sometimes: because when Galileo mean's momentum of speed, he does have a quasi-inertial concept in mind: the value of the speed at that instant, where that speed is considered as a power for the body to move or keep moving a certain amount in a given time.