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."
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
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...