7 Guidobaldo's Marginal Notes in Benedetti's Book

DOI

10.34663/9783945561263-07

Citation

Renn, Jürgen and Damerow, Peter (2012). Guidobaldo’s Marginal Notes in Benedetti’s Book. In: The Equilibrium Controversy: Guidobaldo del Monte’s Critical Notes on the Mechanics of Jordanus and Benedetti and their Historical and Conceptual Backgrounds. Berlin: Max-Planck-Gesellschaft zur Förderung der Wissenschaften.

Guidobaldo's marginal notes in Benedetti's book illustrate a case where different conceptual frameworks are applied to similar mechanical problems and thus are testimony to a competition taking place within the same territory. More generally, these marginal notes reveal the potential tensions and conflicts inherent in the bodies of knowledge carried over from antiquity when these are elaborated, integrated with each other, and applied to the challenging objects of preclassical mechanics (see sections 3.3 and 3.4.3). From an analysis of the marginal notes it becomes clear that Guidobaldo considered Benedetti's entire approach as being misguided. On the one hand, he repeatedly claimed that Benedetti had taken over propositions from his own book on mechanics, such as the assertion that a balance with equal arms would, when deflected from the horizontal position, not return to its original position. On the other hand, he found that the conceptual foundation of Benedetti's approach, based on determining positional heaviness by a perpendicular to the line of inclination, is untenable, leading him to false or problematic claims. In summary, according to Guidobaldo, everything that Benedetti had accomplished was either plagiarized or simply wrong.

The different conceptual foundations of Guidobaldo's and Benedetti's mechanics also account for the different status of certain problems within the theoretical frameworks of their treatises. Thus, while the case of a bent lever and the case of a balance with a weight on one side and a force acting in an arbitrary direction on the other side are rather central to Benedetti, such situations receive at best a marginal treatment in the deductive part of Guidobaldo's treatise. Benedetti's framework was evidently more apt than Guidobaldo's to deal with cases in which force and lever arm are not perpendicular to each other, or more generally, with cases in which the acting force and the path of the motion constrained by a mechanical device are not parallel as is the case, for instance, for the inclined plane. While this potential was not fully exploited in Benedetti's treatise, not least because of the sketchiness of some of his proofs, it became essential to Galileo's more rigorous and systematic treatment of mechanics modeled after that of Guidobaldo, as well as to the theory of motion built upon it (see section 3.10). Guidobaldo's marginal notes on Benedetti's treatise thus not only illustrate the clash of their different perspectives, but also the developmental potential inherent in this clash. They allow us to witness the beginnings of a process in which the heterogeneous conceptual traditions of early modern mechanics were eventually fused to constitute the framework of classical mechanics in the context of a scientific controversy (see section 1.3).

Guidobaldo's notes are presented here in the context of those passages of Benedetti's text to which they refer. Several parts of the marginalia have been deleted by Guidobaldo himself. Other parts have later been cut off by a book binder. Also, we have been unable to read all of his handwriting.1

7.1 First chapter: the general charge of plagiarism

Benedetti started the part of Diversarum speculationum […] liber that deals with mechanics after a short introduction with a first chapter entitled:

De differentia situs brachiorum librae.

On the difference in the position of the arms of a balance.2

It has been mentioned above that Benedetti sets out in this first chapter to explain the idea of positional heaviness by stating:

Omne pondus positum in extremitate alicuius brachii librae maiorem, aut minorem gravitatem habet, pro diversa ratione situs ipsius brachii.

Every weight placed at the end of an arm of a balance has a greater or a lesser heaviness depending on differences in the position of the arm itself.3

The rest of the chapter elaborates on this idea.

In the introduction which precedes this chapter Benedetti claimed that he presents material that has either never been dealt with previously or has not been sufficiently explained. It is to this bold claim that Guidobaldo reacts in his first marginal note, written after the beginning of the first chapter. The note is placed in the right margin of the page at the height where the text of the first chapter begins. It refers to the chapter as a whole, claiming that it had been taken entirely from his own book:

Hoc primum caput to[tum] desumptum est a n[ostro] mechanicorum libr[o] tractatu de lib[ra].

This entire first chapter is taken from our book on mechanics, from the treatise on the balance.

Fig. 7.1: The first marginal note.

Fig. 7.1: The first marginal note.

As this marginal comment suggests, Guidobaldo was evidently convinced that not only had he derived all the relevant theorems in his own book but also addressed what he saw as the problematic character of the concept of positional heaviness in Jordanus, Tartaglia, and Cardano. As Benedetti's approach corresponds to one of the options of Cardano (see section 3.7), he was apparently under the erroneous impression that he had thus dealt with Benedetti's approach as well.

7.2 Second chapter: the neglect of the cosmological context

Benedetti's second chapter is entitled:

De proportione ponderis extremitatis brachii librae in diverso situ ab horizontali.

On the ratio of the weight at the extremity of the arm of a balance in various positions with respect to the horizontal.4

Fig. 7.2: Figure at the beginning of the second chapter.

Fig. 7.2: Figure at the beginning of the second chapter.

The chapter considers, as we have discussed, the changing effect of a weight in different positions of the arm of a balance. Guidobaldo began his marginal note on the lower left margin of the page opening the chapter and continued it at the bottom. The note refers to the concluding sentence on the page:

Unde fit ut hoc modo pondus magis aut minus a centro pendet, aut eidem nititur: atque haec est causa proxima, et per se, qua fit ut unum idemque pondus in uno eodemque medio magis aut minus grave existat.

Hence it results that in this way a weight hangs more or less from the center or is sustained by it. And this is the proximate and essential cause why it happens that one and the same weight in one and the same medium is more or less heavy.5

Fig. 7.3: Marginal note to the second chapter.

Fig. 7.3: Marginal note to the second chapter.

In his comment, Guidobaldo questioned the basic geometrical set-up of Benedetti's argument because it does not take into account that the perpendicular lines are not parallel but have to meet at the center of the world. As we have discussed, in his own book, he had extensively criticized Tartaglia's approach in a similar way – not because of an overEOAemphasis on precision, but for reasons of logical consistency enforced by Tartaglia's conceptualization of oblique descents.

non est neque proxima neque per se; nam [pond]us in F brachii [BF] non est equegrave ut pondus in U brachii BU ; [nec] pondus in E brachii BE est equegrave ut pondus [in] U brachii BU. Unde tota haec demonstratio falsa est.

because that [i.e. the greater or smaller extent to which a weight rests on the center] is neither the next [cause] nor the [cause] by itself. For the weight at F of the arm BF is not equally heavy as the weight U of the arm BU; nor is the weight at E of the arm BE equally heavy as the weight at U of the arm BU. Whence this entire demonstration is false.

Guidobaldo thoroughly examined what he considered to be the problematic foundation of Benedetti's mechanics also in his research notebook, the Meditatiunculae, under the heading:6

Contra Cap. 2 Jo. de Benedicti de Mechanicis

Against chapter 2 of Giovanni Benedetti's [treatise] on Mechanics

Fig. 7.4: In his notebook Guidobaldo reconsidered Benedetti's analysis of the bent lever, conceived as a balance with one horizontal arm BD and an oblique arm in the positions BF or BE. He stressed the difference between Benedetti's treatment and a treatment that takes into account the finite distance of the bent lever from the center of the world. For this purpose, Guidobaldo compared the line LUS parallel to the line AQ, connecting the fulcrum B of the balance with the center of the world, with the line FM connecting the weights on the beam with the center of the world. He concluded that it is the weight at S, at the point where the line LUS meets the circle the beam describes around the fulcrum, and not, as claimed by Benedetti, the lower weight at E, that will be equally heavy as the weight at U.

Fig. 7.4: In his notebook Guidobaldo reconsidered Benedetti's analysis of the bent lever, conceived as a balance with one horizontal arm BD and an oblique arm in the positions BF or BE. He stressed the difference between Benedetti's treatment and a treatment that takes into account the finite distance of the bent lever from the center of the world. For this purpose, Guidobaldo compared the line LUS parallel to the line AQ, connecting the fulcrum B of the balance with the center of the world, with the line FM connecting the weights on the beam with the center of the world. He concluded that it is the weight at S, at the point where the line LUS meets the circle the beam describes around the fulcrum, and not, as claimed by Benedetti, the lower weight at E, that will be equally heavy as the weight at U.

He addressed Benedetti's claims by reconstructing them from the perspective of his own conceptual framework based on the concept of center of gravity. As indicated in his marginal notes, Guidobaldo rejected Benedetti's approach because it supposedly did not take into account the finite distance of the weights from the center of the world and hence the fact that the plumb lines are not parallel to each other.

In his diagram, Guidobaldo compared the line LUS parallel to the line AQ through the fulcrum with the line FM connecting the upper weight F with the center of the world (see figure 7.4). S is the point where the line LUS meets the circle the beam describes around the fulcrum, which is above the position of the lower weight E. He next considered a bent lever made of the oblique arm BS, rigidly connected to the straight arm BD, assuming that BU is half BD.

If now a weight is placed at S which is double the weight at D, the bent lever will be in equilibrium, as Guidobaldo showed with reference to his book, because the center of gravity of the weights at S and at D will be at the point R, which will be in its lowest place on the vertical line BQ.

He then concluded that it is the weight at S, but not the lower weight E, that will be equally heavy as the weight at U. He proceeded to demonstrate this in greater detail by considering the proportions in which the line connecting the two weights of the bent lever is cut by the perpendicular BQ for the two cases, i.e. the weight being placed at S and weight being placed at E.

Guidobaldo concluded that the same weight is heavier at S than at E. He then turned to a closer consideration of the upper weight F. Again he constructed a bent lever LBD in equilibrium in order to compare it with the bent lever formed with the upper weight F. And again he showed that the weight is heavier at L than at F, concluding:

Et quibus etiam constat idem pondus in F, et in U, et in E, diversi modo gravitare. Gravius est enim in situ E quam in U et in F. In U vero gravius, quam in F.

From this it is also clear that the weight at F, at U, and at E gravitates in a different way. It is namely heavier in the position E than it is at U and at F. But at U it is heavier than at F.7

Finally, he summarized in almost the same words as in his marginal comment quoted above:

Veluti quoque falsum est propter filum pondus in E est aequegrave, ut pondus in U brachii BU. Non est igitur haec vera et proxima causa, et per se harum gravitatum. Ut ipse profitetur.

In the same way it is also false that because of the thread the weight at E is equally heavy as the weight at U on the arm BU. This is therefore not the true and next cause, nor the essential [cause] of these gravities. As he himself admits.8

7.3 Third chapter: the pitfalls of determining positional heaviness

The third chapter is entitled:

Quod quantitas cuiuslibet ponderis, aut virtus movens respectu alterius quantitatis cognoscatur beneficio perpendicularium ductarum a centro librae ad lineam inclinationis.

That the magnitude of one given weight or the magnitude of one motive force in comparison with another can be found by means of perpendiculars drawn from the center of the balance to the line of inclination.9

The title summarizes the gist of Benedetti's procedure for determining positional heaviness. Guidobaldo left two comments in the right margin of the page opening the chapter – the first short, the second long and with deletions.

Guidobaldo's first comment refers to the figure on the preceding page of Benedetti's treatise to which in turn the first sentence of chapter 3 refers (see figure 7.2):

Ex iis, quae a nobis hucusque sunt dicta, facile intellegi potest, quod quantitas BU quae fere perpendicularis est a centro B ad lineam FU inclinationis, ea est, quae nos ducit in cognitionem quantitatis virtutis ipsius F in huiusmodi situ, constituens videlicet linea FU cum brachio FB angulum acutum BFU.

From what we have already shown it may easily be understood that the length of BU, which is virtually perpendicular from the center B to the line of inclination FU, is the quantity that enables us to measure the force of F itself in a position of this kind, i.e., a position in which line FU constitutes with arm FB the acute angle BFU.10

The point of Guidobaldo's first short comment is probably the same as that of the preceding comment: to stress that Benedetti's diagram fails to take into account that the vertical lines have to actually converge at the center of the world.

Fig. 7.5: Note at the beginning of the third chapter.

Fig. 7.5: Note at the beginning of the third chapter.

Guidobaldo's first comment reads:

diximus hoc f[iguram] esse hoc mod[o]

We said that this figure is in this way

Fig. 7.6: Benedetti's bent lever with forces acting along the oblique lines AC.

Fig. 7.6: Benedetti's bent lever with forces acting along the oblique lines AC.

Guidobaldo's second comment, beginning in the lower right margin and continuing at the bottom of this page, refers to the interpretation of the figure at the bottom of the page and to the argument beginning with the second sentence of the chapter which reads (see figure 7.6):

Ut hoc tamen melius intelligamus, imaginemur libram BOA fixam in centro O ad cuius extrema sint appensa duo pondera, aut duo virtutes moventes E et C ita tamen quod linea inclinationis E idest BE faciat angulum rectum cum OB in puncto B linea vero inclinationis C idest AC faciat angulum acutum, aut obtusum cum OA in puncto A. Imaginemur ergo lineam OT perpendicularem lineae CA inclinationis […]

To understand this better, let us imagine a balance BOA fixed at its center O, and suppose that at its extremities two weights are attached, or two moving forces, E and C, in such a way that the line of inclination of E, that is, BE, makes a right angle with OB at point B, but the line of inclination of C, that is, AC, makes an acute angle or an obtuse angle with OA at point A. Let us imagine, then, a line OT perpendicular to the line of inclination CA […]11

Guidobaldo's second comment thus refers to Benedetti's analysis of the case of a balance with a weight on one side and a force acting in an oblique direction on the other side:

Fig. 7.7: Note beside drawings of bent levers in the third chapter.

Fig. 7.7: Note beside drawings of bent levers in the third chapter.

si intelligamus p[ondus] in C, ut supponi p[otest] ex verbis ipsius, intelligendum est C[T] quoque consolidatam consolidatis TO […]. Unde si intelligamus C pondus et non movens, falsa est i[ta]que si intelligatur C movens ut homi[…] vera esse pote[st] quod [deleted: non] moveat non esse pondus s[i...] ipse [vero] in sequenti accipiat [hoc atque ponderi?] posse demonstratum quare nihil […] ut patet in 7 cap.

In his duobus cap. fundantur omnes authoris demonstrationes ita ut sunt praecipua mechanicorum fundamenta quorum cognita falsitate omnia rem[oventur.]

If we understand that a weight is at C, as we can assume from his own words, then CT must also be understood as being solid [and connected with] the solid lines TO […] If we hence understand that C is a weight and not moving, [the proposition] is false. If it is understood that C moves as […] of a man, it can be true, since what moves is not a weight. [But] if he himself assumes in the following that [this] can be demonstrated [also for a weight], nothing […] therefore as is evident in chapter 7.

On these two chapters all demonstrations of the author are founded inasmuch as they are the first fundaments of mechanics; once their falsity is recognized, everything is rejected.

This comment illustrates Guidobaldo's difficulties in coping with a subject that was apparently unfamiliar to him. Generalizing from the case of the bent lever treated in chapter 2, Benedetti argued, as we have discussed in section 3.9, that the magnitude of a weight or a force can be found by means of perpendiculars drawn from the center of the balance to the line in which the weight or the force acts, that is, the line of inclination which does not, however, have to be a perpendicular. Now this generalization raised problems for Guidobaldo: must this line of inclination be understood as the solid arm of a bent lever with a weight attached to it at the lower end, thus generating a pull downward to the center of the world? Then Benedetti's conclusion would be wrong. Or can the line of inclination also represent a moving force, for instance, the pull of a man acting on the handles of a wheel? Then Benedetti's conclusion may actually be correct.

In his second marginal comment on this page, as well as on the related page in the Meditatiunculae, which we shall discuss immediately below, we see Guidobaldo struggling with these two possibilities. Apparently Guidobaldo believed that while Benedetti's procedure may be applicable to the case of moving forces, it was certainly false for weights tending to the center of the world. In his marginal comment Guidobaldo referred to chapter 7 of Benedetti's treatise, probably because it served as evidence that Benedetti applied this procedure not only to forces but also to weights. As we have discussed above, Benedetti had criticized Tartaglia and Jordanus in this chapter and offered a new analysis of the behavior of a balance removed from its horizontal position. When taking into account that the lines of inclination of the two weights on the balance have to converge at the center of the earth, Benedetti had come to the conclusion – by applying the incriminated procedure – that their positional heaviness must be different, a conclusion with which Guidobaldo could obviously not agree.

Then, in his final comment concluding the second marginal note, Guidobaldo summarized his conviction that the entire foundation of Bene-

detti's approach, as outlined in the first two chapters of his book, is untenable.

Fig. 7.8: In his notebook, Guidobaldo attempted to refute Benedetti's determination of the positional effect of forces acting in an arbitrary direction under the erroneous assumption that such forces can be replaced by weights. Following Benedetti, he considered a broken bent lever BOAC with fulcrum O. For the case of an acute angle BAC, he showed that this broken bent lever cannot be in equilibrium because its center of gravity S can never fall on the perpendicular line OU through the fulcrum.

Fig. 7.8: In his notebook, Guidobaldo attempted to refute Benedetti's determination of the positional effect of forces acting in an arbitrary direction under the erroneous assumption that such forces can be replaced by weights. Following Benedetti, he considered a broken bent lever BOAC with fulcrum O. For the case of an acute angle BAC, he showed that this broken bent lever cannot be in equilibrium because its center of gravity S can never fall on the perpendicular line OU through the fulcrum.

In his research notebook Guidobaldo dealt at even greater length with the same problem. He began his notes with the following comment:

Falsum est igitur ex dictis, quod in principio tertii capitoli inquit. Praeterea demonstratio falsa quoque videtur.

From what has been said, what he claims in the beginning of the third chapter is therefore false. Moreover also the demonstration appears to be wrong.12

Guidobaldo extensively refuted Benedetti's procedure under the erroneous assumption that the latter had claimed that forces can indiscriminately be replaced by weights. In particular, Guidobaldo considered a broken bent lever BOAC with fulcrum O, weights E and C, straight arm BO, and broken arm OAC, just as Benedetti had discussed it for the two cases of an acute and an obtuse angle BAC (see figures 7.8 and 7.9).

He first recapitulated Benedetti's procedure, assuming that a vertical line OT be drawn from the fulcrum to the line AC representing the oblique end of the bent lever. He stated that when the weight C is instead placed at the end of the horizontal line OI, whose length is the same as that of the perpendicular OT, it will, according to Benedetti, be in equilibrium with the weight E, if the weight C is to the weight E as is BO to OT or OI. Guidobaldo then summarized that Benedetti claimed that also the bent lever formed by the straight arm BO and the oblique arm OTC, where a force represented by the weight C acts along the line TC, will be in equilibrium, which he doubted. Ironically referring to Benedetti's use of the term common science, he wrote:

Fateor me hanc quamdam communem scientiam non intelligere.

I admit that I do not understand this certain common science.13

Guidobaldo reformulated Benedetti's claim by stating that the same weight C will be in equilibrium with the weight E, whether it is placed on the straight balance BOI or on the broken bent lever BOTC. He thus replaced Benedetti's conception of a force acting along an oblique line with that of a weight always tending downward and necessarily arrived at absurd conclusions.

Guidobaldo showed in particular that the same weight will be heavier on the horizontal at the point I than along the bent lever at T, demonstrating that the bent lever TOB will not be in equilibrium if the straight lever BOI is in equilibrium. In order to demonstrate this, he again proceeded by finding the center of gravity of the weights E and C placed at T. More precisely, Guidobaldo determined a position for the weight C in which the bent lever is in equilibrium, a position, however, that is distinct from T, so that it follows that T cannot be the equilibrium position for this weight. For this purpose, he prolongued the line BT to D, just underneath I, so that it is immediately evident that, if the weight C is placed at D, the center of gravity of the two weights will be just underneath the fulcrum.

He then continued to show by the same pattern that also the bent lever BOC cannot be in equilibrium because its center of gravity S can never fall on the perpendicular line OU through the fulcrum. And finally he extended this argument to the broken bent lever BOTC.

Fig. 7.9: Following Benedetti, Guidobaldo considered in his notebook the broken bent lever BOAC also for the case of an obtuse angle BAC, under the assumption that E and C are weights. He again came to the conclusion that their center of gravity S can never fall on the perpendicular line OU through the fulcrum and that hence the lever cannot be in equilibrium.

Fig. 7.9: Following Benedetti, Guidobaldo considered in his notebook the broken bent lever BOAC also for the case of an obtuse angle BAC, under the assumption that E and C are weights. He again came to the conclusion that their center of gravity S can never fall on the perpendicular line OU through the fulcrum and that hence the lever cannot be in equilibrium.

Guidobaldo next addressed the case in which the bent lever is characterized by an obtuse angle BAC, showing that the weight at T has a smaller heaviness than the weight at I (see figure 7.9).

In his concluding remarks, however, he began to waver. He once again stated that Benedetti is completely in error when applying his procedure to weights. But he admitted that it may be when one is dealing with a force:

Falsa igitur est demonstratio. Fallacia vero est, cum inquit, continget, ut BOT communi quadam scientia, non moveatur situ.

Et est omnino falsum si intelligatur C esse pondus, quod in centrum mundi sempre tendit. Ut ipse supponere videtur. Et ut ipse in seguentibus capitolis accipit hoc tamquam de ponderibus demonstratum.

At vero si intelligatur C potentia movens, ut hominis, qui potest trahere T per rectam lineam TC, tunc vera esse potest demonstratio. Ut patet ex tractatum de axe in peritrochio nostrorum Mechanicorum.

The demonstration is therefore false. But the fallacy is, as he says, that it is the case that BOT by some common science does not change its place.

And it is totally false if C is understood to be a weight which always tends to the center of the world, as he seems to assume, and as he in the subsequent chapters assumes it to be demonstrated as if it holds for weights.

But if C is understood to be a moving power, like that of a man who can draw T along the straight line TC, then the demonstration can be true. As is clear from the treatise on the wheel and axle of our [book] on mechanics.14

Remarkably, while the Copernican Benedetti speaks of the center of the region of the elements (centrum regionis elementaris), Guidobaldo insists on the center of the world (centrum mundi). By way of an after-thought, Guidobaldo once again criticized Benedetti's appeal to common science, remarking that this is not worthy of an expert mathematician:

Notandum tamen, quod conclusiones per communem quandam scientiam deductae, non sunt periti mathematici cum propriis uti oporteat.

It nevertheless has to be noted that the conclusions which are inferred by a certain common science are not worthy of an experienced mathematician because he should use his own [demonstrations].15

And by way of a second after-thought, he constructed an extreme case in which it is immeditaly clear that the broken bent lever cannot be in equilibrium if weights are attached to it, rather than forces (see figure 7.10):

Ex hac etiam figura magis patet absurdum, hoc est pondera E C aequeponderare non posse.

From this figure it appears even more absurd, that is, that the weights E C cannot be in equilibrium.16

Fig. 7.10: In his notebook, Guidobaldo concluded his alleged refutation of Benedetti's treatment of the broken bent lever with the construction of an extreme situation in which the two weights E and C are found on the same side of the fulcrum O so that it is obvious that the lever cannot be in equilibrium.

Fig. 7.10: In his notebook, Guidobaldo concluded his alleged refutation of Benedetti's treatment of the broken bent lever with the construction of an extreme situation in which the two weights E and C are found on the same side of the fulcrum O so that it is obvious that the lever cannot be in equilibrium.

7.4 Fourth chapter: on the problem of the material beam

The fourth chapter is entitled:

Quemadmodum ex supra dictis causis omnes staterarum et vectium causae dependeant.

How all causes operating on steelyards and levers depend on the aforesaid causes.17

The chapter deals with the fact that the beam of a balance is not a mathematical line but a material body. Benedetti made use of his earlier treatment of the bent lever to take into account the fact that the weights attached to such a material beam do not act along a beam that can be idealized as a horizontal line, but along oblique lines from the fulcrum to the points of suspension of the weights, which are assumed to be placed at the upper part of the material beam. More specifically, Benedetti stated that if two equal weights are attached to the longer and the shorter arm of the balance, the weight attached to the longer arm will overpower the one attached to the shorter arm. He claimed that such an analysis of the material beam has never been dealt with before, a point that Guidobaldo rejected in his notes.

Fig. 7.11: Left note to the first paragraph of the fourth chapter.

Fig. 7.11: Left note to the first paragraph of the fourth chapter.

Guidobaldo left three marginal comments on the page opening the chapter; in addition he referred in a short note in the right margin to a sentence he underlined. The short comment in the left margin includes a drawing by Guidobaldo. The longest comment begins in the lower half of the left margin and is continued at the bottom of the page; it also comprises a drawing. About half of this comment was deleted by Guidobaldo himself; at least one line has later been cut off.

Guidobaldo's first comment reads (see figure 7.14):

[opo]rtet NU esse [hor]izonti equidistantem [ali]ter quidem unde [vo]let demonstrandum

It is necessary that NU is equidistant from the horizon, differently from [the way] in which he wanted [it] to be demonstrated.

The hand-drawn diagram in the left margin was evidently added in the same context.

Fig. 7.12: Drawing of a material beam at the left side of the first paragraph of the fourth chapter.

Fig. 7.12: Drawing of a material beam at the left side of the first paragraph of the fourth chapter.

The meaning of Guidobaldo's first comment is not entirely clear. It seems to pinpoint the fact that Benedetti designated the upper part of the beam as being horizontal, while, according to Guidobaldo, this is in contrast to what has to be demonstrated. Possibly he referred to Benedetti's own later generalization of his argument from balances to levers in the penultimate sentence of the chapter:

In stateris, recte et proprie appelari potest XIS aut NOU orizontalis, sed in omnium vectium specie, hoc tantum per quandam similtudinem dicatur.

In balances with unequal arms, XIS or NOU can be rightly and properly called horizontal, but in the case of all levers this can be said only with a certain approximation.18

Or Guidobaldo wanted to express that this premise implicitly assumes the weights are connected by a horizontal line, in contrast to Benedetti's own detailed analysis which makes reference to oblique lines according to which the weights supposedly act. To stress this point he may have added the diagram in the margin showing a balance with equal arms.

In any case, Guidobaldo's first comment, his drawing, and also his second comment all refer to the set-up of Benedetti's demonstration in the introduction of the chapter:

Positis igitur duobus ponderibus aequalibus in extremitatibus brachiorum, experientia innotescit, quod pondus ad US appensum, violentiam faciet ponderi appenso ad NX sed nos volumus investigare causam huius effectus, quae a nemine unquam literarum monumentis, quod sciam, consignata fuit.

Now if two equal weights are placed at the ends of the arms, it is clear from experience that the weights appended at US will overpower the weight appended at NX. But we wish to investigate the cause of this effect, which cause has never, so far as I know, been assigned by anyone in the annals of literature.19

The last claim is underlined by Guidobaldo as the point of reference of his second note, in the right margin of this page.

Fig. 7.13: Underlined text with marginal note in the fourth chapter.

Fig. 7.13: Underlined text with marginal note in the fourth chapter.

Underlined text:

quae a nemine unquam literarum monumentis, quod sciam, consignata fuit.

which was never, so far as I know, documented by anybody in the annals of literature.20

Marginal note:

nos in tractatu de vecte propos. XV in libro me[chanicorum]

We [did] in the treatise on the lever, prop. 15, in the book on mechanics.

Guidobaldo thus rejected Benedetti's claim to originality and referred to his own work on mechanics, and in particular to proposition 15 of the part on the lever:

Problema.

Quia vero dum pondera vecte mouentur, vectis quoque grauitatem habet, cuius nulla hactenus mentio facta est: idcirco primum quomodo inueniatur potentia, quae in dato puncto datum vectem, cuius fulcimentum sit quoque datum, sustineat, ostendamus.

Problem.

But since in moving weights with a lever, the lever also has weight, which has not been mentioned up to this point, we shall demonstrate how to find the power which will sustain the lever in a given point, the fulcrum being likewise given.21

As becomes clear from the proof of this proposition, Guidobaldo considered the centers of gravity of the two parts of the material beam, as they are divided by the fulcrum of the lever, and treats the entire beam of the lever as being represented by two weights suspended at the distances of these centers of gravity from the fulcrum. Not only is his procedure entirely different from that of Benedetti. Guidobaldo's and Benedetti's conceptual frameworks actually capture different aspects of the material beam. While Guidobaldo managed to take into account the weight of the beam, Benedetti only dealt with its geometrical extension and focused on the direction of the pull of the attached weight, corresponding in modern terms to the torque of the applied force.

The third comment, also in the left margin of this page, refers to Benedetti's construction of the lines according to which weights act within a material balance. It is accompanied by another diagram of Guidobaldo's to which he referred in the last part of this commentary.

Fig. 7.14: Long note to Benedetti's fourth chapter.

Fig. 7.14: Long note to Benedetti's fourth chapter.

[cu]m pondera pendeant [S]X estque XIS recta linea huius […] quaerenda a lineis IN IU, ut in […]?, quae quidem sint pro[…] immaginariae. Deinde [poste]a facit mentionem [de gra]vitate vectis, et considerat matem[atice] […] ex hac causa ut infra duo vectes

[large passage deleted]

AIU absque OB et pondera at XS, pondus in S maiorem habebit vim, supra pondus in X, vecte AIU, quod idem pondus in S ad pondus in X toto vecte […] [line below cut]

Since the weights hang from SX and XIS is a straight line whose […] is to be found from the lines IN IU, as in […], which are though only […] imaginary. Finally he later mentions the heaviness of the lever and considers in a mathematical way […] from which cause as the two levers below

[large passage deleted]

AIU if without OB and the weights in XS, the weight at S will have more power over the weight at X, by the lever AIU, than the same weight at S to the weight at X with the entire lever […]

Although much of this commentary remains illegible, two salient points of Guidobaldo do emerge: He considered the oblique lines along which, according to Benedetti, the weights attached to a material beam act as being purely imaginary. And he apparently attempted to construct a contradiction within Benedetti's framework by considering the weights being attached to different heights of the material beam. Guidobaldo's drawing shows indeed a beam of approximately twice the height of the original one, with the original one inscribed. In the last legible line of his note he considered the weight on the longer right-hand side of the balance being attached from the original height of the beam, while the weight on the shorter left-hand side is suspended from the beam with double height. Guidobaldo concluded with an argument that he evidently later rejected that, in this constellation, the weight on the longer right-hand side has more power than if it were suspended from the same height as the weight on the left-hand side.

7.5 Fifth chapter: on the problem of the material lever

The fifth chapter is entitled:

De quibusdam rebus animadversione dignis.

On certain facts worthy of notice.22

The chapter deals with levers whose fulcrum is at one end of the lever, while the weight to be lifted by a force acting on the other end is positioned between these ends and somewhere near to the fulcrum. As we have discussed before, Benedetti treated the material lever not with regard to the weight of the beam but only with regard to its geometrical configuration. He hence imagined a rectangular cross-section of such a lever with a weight being placed on top of the beam. One lower corner serves as the fulcrum, the other corner is lifted by the hand. The question then is how the weight exerts a pressure on the corner where the hand is acting. Benedetti claimed that the ratio between that part of the weight that rests on the fulcrum and that part of the weight that rests on the corner where the force is acting is given by the inverse ratio of the horizontal distances of the weight from these two points (see figure 7.15):

Si vero eadem resistentia posita erit in U clarum quoque erit, quod minor pars ponderis N annitetur ipsi U quam ipsi O cum dicta NI a centro U longius quam a centro O distet, et proportio partis ponderis N in O ad proportionem partis ponderis N in U non erit secundum proportionem angulorum UNI et ONI sed secundum proportionem UI ad IO [this proportion is underlined] quod clare comprehendi potest ab huius effectus converso […]

And if the same resistance is placed at U, it will also be clear that a smaller part of weight N will press on U than on O, since NI is farther distant from fulcrum U than from fulcrum O. And the ratio of the part of weight N that rests on O to the part of weight N that rests on U will be equal, not to the ratio of angle UNI to angle ONI, but to the ratio of UI to IO. This may be clearly understood from the converse of this effect […]23

Fig. 7.15: Figure and marginal note in the fifth chapter.

Fig. 7.15: Figure and marginal note in the fifth chapter.

In the sequel Benedetti justified his claim by interpreting the situation of the lever according to the model of a balance suspended from the point where the weight is positioned, with the two lower corners now representing weights. From his procedure of determining effective lever arms by horizontal projection his proposition then followed.

Guidobaldo left a marginal note at the bottom right of page 145 and underlined the letters in Benedetti's text referring to the proportion of lengths in the diagram to which his comment refers:

tandem post mu[lta] veritate coactus dixit proportionem p[artium] ponderis esse secundum OI, IU quod nos in 3 coroll. secundae propositionis de vecte omnia di[ximus].

Finally he said after many [other things], forced by the truth, that the proportion of the parts of the weight is as OI, IU which we have said all in the third corollary of the second proposition about the lever.

Guidobaldo's marginal comment refers to the text at the bottom of the page quoted above which in turn refers to the diagram on the same page. Guidobaldo referred to his own treatment of levers in his book on mechanics, where he also dealt with a lever sustained at its two ends carrying a weight in the middle. He explicitly referred to the third corollary of the second proposition about the lever which reads:

Ex hoc quoque elici potest, si duae fuerint potentiae, una in A, altera in B, et utraque sustentet pondus E; potentiam in A ad potentiam in B esse, ut BC ad CA.

From this likewise it may be deduced that, if there are two powers, one at A and the other at B, and both sustain the weight E [suspended from point C], the power at A will be to the power at B as BC is to CA.24

He also justified his claim by exchanging the roles of fulcrum and force, but he did not take into account any directional effects of these forces, considering the lever without extension.

7.6 Seventh chapter: on the core question of the equilibrium controversy

The seventh chapter is entitled:

De quibus erroribus Nicolai Tartaleae circa pondera corporum et eorum motus, quorum aliqui desumpti fuerunt a Jordano scriptore quodam antico.

On certain errors of Niccolò Tartaglia on the weights of bodies and their motions, some taken from a certain ancient writer Jordanus.25

In this chapter Benedetti criticized Tartaglia's account of the variation of the positional heaviness of a body on a balance changing its position. As we have discussed extensively above (see section 6.2), he rejected, in particular, Tartaglia's claim that a balance would return to its original horizontal position because the weight that has moved upward becomes positionally heavier, while the weight that has moved downward becomes positionally lighter. Benedetti first pointed out that Tartaglia should not have compared the descents of the two weights but the descent of one weight with the ascent of another. In his annotations Guidobaldo did not fail to notice that, in his own book, he had already drawn attention to this circumstance. Then Benedetti reconsidered, as we have also discussed, the entire situation from a cosmological perspective, concluding that the weight that has moved upward actually becomes positionally lighter, while the weight that has moved downward becomes positionally heavier. Benedetti's argument is based on his procedure of determining effective lever arms by drawing perpendiculars to the lines of inclination of the two weights (see section 3.9). It is remarkable that his first criticism seems to suggest, in agreement with Guidobaldo's opinion, an indifferent equilibrium of the balance (indeed, under terrestrial circumstances it necessarily leads to that conclusion). In contrast, his second criticism (taking into account the cosmological perspective) implies that the balance would actually proceed to the vertical. Benedetti did not, however, actually make this explicit.

Page 148 has two comments by Guidobaldo, one in the middle of the page in the left margin, the other further below, also beginning in the left margin and continuing at the bottom of the page; the latter comment refers to a line in Benedetti's text underlined by Guidobaldo. His two marginal notes address the ambiguity of Benedetti's text. They end in the definitive rejection of Benedetti's method of determining positional heaviness, which – in the eyes of Guidobaldo – is in conflict with his fundamental insight into the indifferent nature of the equilibrium of a balance.

Fig. 7.16: Marginal note to Benedetti's seventh chapter.

Fig. 7.16: Marginal note to Benedetti's seventh chapter.

Guidobaldo's first comment reads:

desumptum est ex iis [quae] dicta sunt a nobis [in] tractatu de vecte […] supponit pondera non moveri ut [re] vera est. Serius concludit oppositum.

This is taken from what has been said by us in the treatise on the lever. He assumes that the weights do not move which is true. Later he concludes the opposite.

The comment refers to the passage in which Benedetti pointed out that the descent of one weight should be compared to the ascent of the other (see figure 3.11):

Sed in secunda parte quintae propositionis non videt quod vigore situs eo modo, quo ipse disputat, nulla elicitur ponderis differentia. Quia si corpus B descendere debet per arcum IL corpus A ascendere debet per arcum US aequalem, et similem eadem quoque rationem situatum, ut est arcus IL unde ut est facile corpori B descendere per arcum IL difficile ita erit corpori A ascendere per arcum US. Haec autem quinta propositio Tartalea est secunda quaestio a Iordano proposita.

And in the second part of the fifth proposition, he fails to see that no difference in weight is produced by virtue of position in the way in which he argues. For if body B must descend on arc IL, body A must ascend on arc US, equal and similar to arc IL and placed in the same way. Therefore, just as it is easy for body B to descend on arc IL, it will be difficult for body A to ascend on arc US. And this fifth proposition is the second question proposed by Jordanus.26

Guidobaldo's second note at the bottom of the page refers to the diagram on the following page 149 (see figure 7.17).

Fig. 7.17: Drawing of a balance in a cosmological context in the seventh chapter.

Fig. 7.17: Drawing of a balance in a cosmological context in the seventh chapter.

It also refers to a passage in the text on page 148 that has been underlined by Guidobaldo:

Pondus igitur ipsius A in huismodi situ, pondere ipsius B gravius erit.

Therefore the weight of A in this position will be heavier than the weight of B.27

This is the conclusion of Benedetti's consideration of two weights on a balance in an oblique position from a cosmological perspective, amounting to the statement that the weight A that has been lowered has become positionally heavier than the weight B that has been lifted. Guidobaldo's second comment reads:

[suppo]nit pondera AB non moveri. Hac demonstratione pondus A gra[vius] [e]st pondere B quia haec gravi[tates] metiuntur ex lineis perpendicularibus OT, OE quarum OT maior est, sequitur pondus A in hoc situ gravius esse pondere B in hoc situ. Dico igitur quod subterfugiet [p]ondus A deorsum non moveatur et B sursum? Libra ergo AB non manebit ut supposuit, et ut re vera manet. [Qua]re si volens errores Iordani et Tartaleae (quorum errorum nec solvit contradictiones) incidit, et si non in peiora, tamen in aequalia [abs]urda. Unde perspicuum est, quod sit inanis, et falsa haec consideratio suis perpendicularibus facta, quam […]

He assumes that the weights AB are not moved. By this demonstration A is heavier than weight B, because these gravities are measured by the perpendicular lines OT, OE, from which OT is the greater, it thus follows that weight A is in this position heavier than weight B in this position. I therefore say what escapes him: does not weight A move downward and weight B upward? The balance AB will therefore not remain as he assumes and as it truly remains. If he therefore willingly cuts into the errors of Jordanus and Tartaglia (whose contradictions he does not resolve), and if he does not make them worse then nevertheless to an equal extent absurd. From which it is evident that this consideration of his which he makes about the perpendiculars is empty and false as [line cut off]

Fig. 7.18: Bottom note of the seventh chapter.

Fig. 7.18: Bottom note of the seventh chapter.

As we have discussed above, Guidobaldo had, in his own book, similarly considered the case of a balance in an oblique position from a cosmological perspective, also using the concept of positional heaviness.28 He had arrived at the conclusion, in agreement with his general conviction, that the two weights on such a balance are equally heavy positionally. Against this background, Benedetti's method of determining positional heaviness, necessarily in contradiction with this conclusion, must have appeared entirely unacceptable to Guidobaldo as he indeed clearly stated in this marginal note.

7.7 Eighth chapter: on plagiarizing the criticism of Jordanus and Tartaglia

The eighth chapter is entitled:

Quod autem idem Tartalea in 6. propositione, et Iordanus in secunda parte secundae propositionis scribunt, maximum quoque errorem in se continet.

What Tartaglia writes in proposition 6 and Jordanus in the second part of proposition 2 also contains a most serious error.29

Fig. 7.19: Drawing of the convergence of perpendiculars in the eighth chapter.

Fig. 7.19: Drawing of the convergence of perpendiculars in the eighth chapter.

In the beginning of this chapter Benedetti criticized, as we have discussed, Tartaglia's way of determining positional heaviness by means of angles of contact between the curved path of a weight and a perpendicular. He showed that this procedure leads to a contradiction when the convergence of these perpendiculars at the center of the world is taken into account, as he indicates in his drawing. Benedetti concluded his analysis with a general rejection of the method of Tartaglia and Jordanus.

Omnis autem error in quem Tartalea, Iordanusque lapsi fuerunt ab eo, quod lineas inclinationum pro parallelis vicissim sumpserunt, emanuit.

Now the whole error into which Tartaglia and Jordanus fell arose from the fact that they took the lines of inclination as parallel to each other.30

Fig. 7.20: Marginal note in the eighth chapter.

Fig. 7.20: Marginal note in the eighth chapter.

Benedetti's argument is strikingly similar to those of Guidobaldo against this method, as we have discussed above. Page 150 has a single short comment by Guidobaldo in the middle of the left margin, next to a text passage in which Benedetti finally rejected the method of Tartaglia and Jordanus. In his marginal comment Guidobaldo pointed to the fact that Benedetti's argument, in his view, has been taken from his own book:

[ex] meo tractatu [de lib]ra

from my treatise on the balance

7.8 Tenth chapter: on Aristotle and the composition of motions

The tenth chapter is entitled:

Quod linea circularis non habeat concavum cum convexo coniunctum, et quod Aristoteles circa proportiones motuum aberraverit.

That the circumference of a circle does not have a concavity joined with a convexity, and that Aristotle was mistaken in the ratios of motions.31

The chapter deals with the introductory part of the Aristotelian Mechanical Problems in which the curious properties of the circle and the composition of motions are treated. Benedetti disputed the claim of the Aristotelian author that the circle seems to unite the convex with the concave, essentially arguing that one should distinguish between the circular surface included by the circumference and the plane with a circular hole that is also delimited by that circumference. Guidobaldo left two rather long comments in the left margin of this page. In his first marginal note Guidobaldo rejected Benedetti's distinction because its application would require, according to him, an intervening space which, however, is not given. In his discussion of the Aristotelian analysis of the composition of motions Benedetti questioned the alledged claim of the Aristotelian author that if a body moves along a given line, it moves according to one definite proportion rather than according to another one. In particular, Aristotle maintained that when an object moves along the diagonal it will always move in the ratio of the sides of the parallelogram.32 Benedetti showed instead that the same trajectory can be generated by motions following different proportions. In his second marginal comment Guidobaldo pointed to the fact that Benedetti failed to understand Aristotle's argument and that his objection is therefore irrelevant. Guidobaldo stressed in this comment that what matters to Aristotle is only the fact that, when a body is moved in a fixed ratio it necessarily travels in a straight line, independently from the fact that the same straight line may be traversed also by a motion that is given by a different ratio.

More specifically, Guidobaldo's first comment refers to the beginning of the chapter:

Aristoteles in principio quaestionum Mechanicorum ait lineam, quae terminat circulum videtur convexum habere coniunctum cum concavo, quod falsum est: quia huismodi linea partes nullas secundum latitudinem habet, (ut ipse etiam confirmat) sed est idem convexum circuli: linea vero quae terminus est superficiae ambientis, et amplectentis circulum est eadem concavitas dictae superficiae eundem circulum ambientis, quae nullam convexitatem habet et haec duae sunt lineae, quarum una diversa est ab alia, neque altera alterius, quod ad convexum, et ad concavum attinet.

Aristotle at the beginning of Questions of Mechanics says that the line which bounds the circle seems to unite the convex with the concave. But this is false. For a line of this kind has no thickness (as Aristotle himself also asserts), but is identical with the convex boundary of the circle. On the other hand, the line that bounds the surrounding surface and encloses the circle is identical with the concavity of the surface that surrounds the circle, a surface which has no convexity. And these are two lines of which one is different from the other, and not part of the other, so far as pertains to convexity and concavity.33

Fig. 7.21: First marginal note in the tenth chapter.

Fig. 7.21: First marginal note in the tenth chapter.

Guidobaldo commented:

intellexit Aristotelem loqui [quod li]nea circumferentiae area dabitur; duae linee se tangunt secundum latitudi[nem], et sunt invicem separatae. [Hoc] fieri non potest. Imo inter [ea]s cadet superficies quaem[ad]modum inter duo puncta [s]it linea

He understood that Aristotle claimed that the line of the circumference will be given by the area. Two lines touch each other along their latitude and are separate from each other; this cannot be. Therefore a surface falls in between just as there is a line between two points

Guidobaldo's second comment refers to Benedetti's criticism of Aristotle's proof of the composition of motions and to the figure on page 152 (see figure 7.22).

Fig. 7.22: The composition of motions discussed in the tenth chapter.

Fig. 7.22: The composition of motions discussed in the tenth chapter.

Benedetti wrote:

Cui respondeo, punctum A quod movetur in linea AM ab A versus M usque ad F non moveri ab aliqua proportione determinata magis quam ab alia: unde non solum possumus imaginari dictum punctum A moveri ab A usque ad F eiusdem velocitatis sub alia quadam proportione, sed etiam sub alia, quae iam datae contraria sit, ut est proportio ipsius AC ad AB imaginantes moveri A versus C et AC versus BM delatam. […] Huiusmodi igitur consideratio ab Aristotele facta, nullius est momenti.

To Aristotle I reply that the fact that point A moves on line AM from A towards M as far as F does not mean that it moves according to one definite proportion rather than some other. Thus we can suppose that point A moves from A to F not only according to one ratio of the same velocity, but also some other which is the very opposite of the first ratio – e.g. the ratio of AC to AB, it being imagined that A moves toward C and AC toward BM. […] Hence the discussion on this point by Aristotle is of no value.34

Fig. 7.23: Second marginal note in the tenth chapter.

Fig. 7.23: Second marginal note in the tenth chapter.

Guidobaldo commented:

[ho]c non intelligit [demonstr]ationis Aristotelis. Nam [paru]it Aristoteli quando ali[quid] [m]ovetur secundum aliquam propor[tione]m, illud quidem moveri [secundum r]ectam lineam, quod si sup[ponere ea]m lineam secundum alias proportiones moveri potest, [n]ihil interest [se]d haec nullius sunt [mom]enti

He does not understand that part of Aristotle's proof. In fact, Aristotle held that, if something is moved according to some proportion, it will surely move according to a straight line; but, if it is assumed that the line can move according to other proportions, that makes no difference, but these things are of no value.

7.9 Twelfth chapter: saving Aristotle in the equilibrium controversy

The twelfth chapter is entitled:

De vera causa secundae, et tertiae quaestionis mechanicae ab Aristotele non perspecta.

On the true cause not perceived by Aristotle of the Mechanical Questions 2 and 3.35

Fig. 7.24: Drawing of a balance in an oblique position in the twelfth chapter.

Fig. 7.24: Drawing of a balance in an oblique position in the twelfth chapter.

The chapter deals with the question of how a balance either supported from above or from below behaves when it is removed from the horizontal position (see figure 7.24). For the case in which the balance is supported from above Benedetti agreed with the Aristotelian conclusion that it will return to the horizontal position, but justified this behavior with his technique for determining positional heaviness. For the case in which it is supported from below he disagreed with Aristotle who seemed to suggest that the balance will stay in its position. On this point Guidobaldo was of the same opinion, as the following quotation from his book on mechanics shows:

Nam cum in secunda parte secundae quaestionis proponit, cur libra, trutina deorsum constituta, quando deorsum lato pondere quispiam id amouet, non ascendit, sed manet? non asserit adhuc libram deorsum moueri; sed manere. Quod in vltima quoque conclusione colligisse videtur.

For in the second part of the second question he asks, ‘Why, when the support is below, the balance being carried downward and released, it does not rise again, but remains?' Here he affirms not that the balance moves downward, but that it remains, which he seems to have deduced in the last conclusion.36

In contrast to Benedetti, Guidobaldo was convinced, however, that Aristotle's position can be defended, which is also the point of his marginal comment and in line with his appreciation of the ancient heritage, including the Aristotelian work on mechanics. In his own book he argued in fact that the balance does not move further downward because it is prevented from doing so by the support on which it rests.

Fig. 7.25: Marginal note to Benedetti's twelfth chapter.

Fig. 7.25: Marginal note to Benedetti's twelfth chapter.

Guidobaldo's comment refers to the following text at the lower part of the page:

In secunda deinde huius quaestionis parte, in qui scribit libram in situ, in quo posita est, firmam manere, toto coelo aberrat, quia necessarium est, ut omnino cadat, eousque quo spartum sursum remaneat: ablato tamen omni impedimento, quod nulla eget probatione, cum natura sua clarissime pateat.

Then in the second part of this problem, in which Aristotle writes that a balance remains fixed in the position in which it has been placed, he is completely mistaken. For it must continue to fall until the support remains above it, with the assumption, however, that all impediment to this is removed. This proposition requires no proof, since by its own nature it is perfectly clear.37

Guidobaldo's comment reads:

[Aristotel]es potest defendi ut [in] tractatu de libra [a n]obis factum fuit

Aristotle can be defended as it was done by us in the treatise on the balance

7.10 Fourteenth chapter: Aristotle's wheel and the problem of infinite limits

The fourteenth chapter is entitled:

Quod rationes ab Aristotele de octava quaestione conficta sufficientes.

That the reasons by Aristotle in Questions of Mechanics 8 are not adequate.38

Fig. 7.26: The motion of a polygonal shape discussed in the fourteenth chapter of Benedetti's book.

Fig. 7.26: The motion of a polygonal shape discussed in the fourteenth chapter of Benedetti's book.

This chapter also deals with the Aristotelian Mechanical Problems, here with the question of why bodies of circular shape are easier to roll than others.39 Benedetti considered various rotational motions, the rotation of carriage wheels, of pulley wheels, and of potter's wheels. He compared the motion of a wheel with that of a polygon and gave reasons why the motion of the former is easier than that of the latter (see figure 7.26). He argued, for instance, that, when a polygon is rolled along a plane, its center will go up and down, its upward motion will require an effort, while the center of a wheel will always maintain the same distance from the center, that is, as he formulated, from the goal of heavy bodies. He considered the circular shape as the limiting case of polygonal shapes with ever more angles. One short comment by Guidobaldo is found in the left margin of this page. In his comment, Guidobaldo expressed his skepticism about this limiting process.

His note refers to the following passage of Benedetti's argument:

Si ergo quanto plures angolos habebit dicta figura, tanto ad circunvolvendum hoc modo agilior erit. Circularis igitur figura, quae ex infinitis angulis efficitur, omnium agillima erit.

The more angles the said figure will have, therefore the more suitable it will be to rotate in this way. Hence the circular shape, which is constituted from infinite angles, is the most suitable of all.40

Fig. 7.27: Marginal note to the fourteenth chapter of Benedetti's book.

Fig. 7.27: Marginal note to the fourteenth chapter of Benedetti's book.

Guidobaldo noted:

circularem ex infinitis [ang]ulis constare fateor ignotum esse

I confess that it is unknown [to me] that the circular is composed from infinite angles

7.11 Sixteenth chapter: on Aristotle's empty balance

The sixteenth chapter is entitled:

Quod Aristotelis rationes de decima quaestione sint reiiciendae.

That Aristotle's explanation of Questions of Mechanics 10 must be rejected.41

The chapter deals with another topic of the Aristotelian Mechanical Problems, the greater readiness of an empty balance to move. Benedetti approached the subject by comparing two balances, one carrying two small weights, the other two large weights (see figure 7.28). Now according to him Aristotle wonders about the fact that the balance with the smaller weights moves more rapidly when on one of its arms another small weight is placed than when the same small weight is placed on one of the arms of the balance with the large weights. Benedetti essentially argued that Aristotle would have no reason to wonder had he appropriately taken into account his own dynamical principles (see section 3.4.1).

Benedetti explained:

quia semper ineunda est ratio proportionis virtutis mouentis super mobile; quod ipse non fecit.

For the ratio of the moving force to the body moved must always be considered; and Aristotle did not do this.42

Fig. 7.28: Drawing of two balances, one carrying small weights, the other large weights, as discussed in the sixteenth chapter of Benedetti's book.

Fig. 7.28: Drawing of two balances, one carrying small weights, the other large weights, as discussed in the sixteenth chapter of Benedetti's book.

In his short marginal note Guidobaldo seems to express his surprise at Benedetti's claim that Aristotle was wondering, apparently incapable of giving an adequate solution to a problem he had posed himself. The comment refers to the following passage of Benedetti's text (see figure 7.28):

Sit exempli gratia libra AIE quae in utraque extremitate unciam unam solam ponderis obtineat, et sit libra NIU aequalis priori, quae pro singula extremitate unam ponderis libram habeat. Aristoteles admiratur, quod addendo ipsi E mediam ponderis unciam, brachium IE velocius cadat, quam adiiciendo ipsam mediam unciam ipsi U brachii IU.

Let there be, for example, a scale AIE which has at the extremity of each arm merely one ounce of weight; and let there also be a scale NIU, exactly like the former one, which has one pound of weight on each end. Aristotle wonders about the fact that, when he adds a half-ounce weight at E, arm IE falls more rapidly than when he adds that same half-ounce at U, the extremity of arm IU.43

Fig. 7.29: Marginal note to the sixteenth chapter of Benedetti's book.

Fig. 7.29: Marginal note to the sixteenth chapter of Benedetti's book.

Guidobaldo noted:

quod admiratur Aristoteles

what was Aristotle wondering about

7.12 Twentieth chapter: on reducing the wedge to the lever

The twentieth chapter is entitled:

De vera ratione 17 quaestionis.

On the true explanation of question 17.44

This chapter also deals with the Aristotelian Mechanical Problems, here with the question of how the wedge is to be treated according to the model of the lever. Benedetti argued that Aristotle failed to properly reduce the wedge to the lever:

Decimaseptima quaestio ab Aristotele haud benè percepta fuit, quia is non accommodat partes vectis suis locis.

Question 17 was not correctly understood by Aristotle, for he did not assign the parts of the lever to their correct places.45

Fig. 7.30: A comparison of two levers, one with the fulcrum in the middle, the other with the fulcrum at one end, as discussed in the twentieth chapter of Benedetti's book.

Fig. 7.30: A comparison of two levers, one with the fulcrum in the middle, the other with the fulcrum at one end, as discussed in the twentieth chapter of Benedetti's book.

In order to improve on Aristotle Benedetti began his discussion by comparing two levers, one with the fulcrum in the middle, the weight at one and the force at the other end, the other lever having the fulcrum at one end, the weight in the middle and the force at the other end (see figure 7.30). It is the latter kind of lever that he applied to analyze the wedge, but it first had be reduced to the ordinary lever with the fulcrum in the middle. He argued in fact that when the weights, the distances between weight and fulcrum, and the distances between force and fulcrum are equal in the two levers, a force sufficient to raise the weight with one lever will also be sufficient to raise the weight with the other lever. By way of justification he referred, first of all, to common science (scientia communis), and then to his principles treated in chapters 4 and 5. In his marginal note Guidobaldo criticized Benedetti for his all too generous use of the reference to common science (scientia communis).46

More specifically, his comment refers to the passage:

Et quia omnia supponuntur aequalia, clarum quoque erit, communi scientia, tantam virtutem in N quanta sufficiet ad attollendum A in U quoque suffecturam ad elevandum E oportebit attollere U.

And because all are assumed equal, it will also be clear, by common science, that the force at N required to raise A will also be sufficient at U to raise E.47

Guidobaldo wrote in the upper left margin of this page:

[…] sua communis [scient]ia multa probat [si]ne demonstratione

His common science demonstrates much without proof

Fig. 7.31: Marginal note to the twentieth chapter of Benedetti's book.

Fig. 7.31: Marginal note to the twentieth chapter of Benedetti's book.

7.13 Twenty-first chapter: the plagiarized pulley

The twenty-first chapter is entitled:

De vera et intrinseca causa trochlearum.

On the true and intrinsic explanation of compound pulleys.48

Fig. 7.32: Reducing the pulley to the lever, as discussed in the twenty-first chapter of Benedetti's book.

Fig. 7.32: Reducing the pulley to the lever, as discussed in the twenty-first chapter of Benedetti's book.

The chapter deals with the explanation of the pulley, and in particular with the way it can be reduced to the lever or the balance (see figure 7.32). In the middle of the right margin of page 163, Guidobaldo left a short comment. Another related comment is found on the subsequent page. Guidobaldo's first comment refers to the passage in the middle of page 163.

Imaginemur separatim stateram GH cuius centrum sit K ita situm, ut brachium GK sit duplum ad brachium KH supponendo igitur in puncto G pondus aut virtutem moventem unius librae, et in H duarum librarum, absque dubio haec duae virtutes in huismodi distantiis a centro aequales invicem erunt, ob rationes prioribus capitibus iam allatas, et statera orizontalis manebit.

Let us consider, separately from the preceding figure, a balance GH with fulcrum K so situated that arm GK is double the arm KH. Now if we assume a weight or moving force of one pound at point G, and of two pounds at H, clearly these two forces at these distances from the center will be equal to each other for the reasons already set forth in previous chapters, and the scale will remain horizontal.49

In his two comments on this chapter Guidobaldo is, in a sense, less critical of Benedetti than in his other notes. He remarked, however, that Benedetti should have referred to Aristotle when mentioning the law of the lever in his first comment instead of referring to his own work.

Fig. 7.33: First marginal note to the twenty-first chapter of Benedetti's book.

Fig. 7.33: First marginal note to the twenty-first chapter of Benedetti's book.

His first comment reads:

hoc non [con]dit auc[tor] sed Aristoteles

This foundation is not laid by the author but by Aristotle

In the second comment to this chapter – in the lower left margin of the next page – Guidobaldo criticized Benedetti for the lack of acknowledgement that his own treatment of the pulley receives. He accused him of wrongly pretending to add something new when dealing with the compound pulley and its reduction to the balance. Benedetti mentally replaced the compound pulley with a sequence of connected balances, arriving at the conclusion that for a pulley with four wheels, a force that amounts to one fourth suffices to lift a given weight. Up to this point Guidobaldo agreed with him and admitted that this chapter is rather clear, even if Benedetti added unnecessary complications. In Guidobaldo's eyes what is true comes anyway from his own work on the pulley which, however, is not mentioned by Benedetti.

More specifically, the comment on the lower part of page 164 refers to the following passage of Benedetti's text and to the figure on page 163 (see figure 7.32):

Hucusque scientifice novimus pondus, aut virtutem ipsius S quae est dimidium ipsi I sustinere vim ipsorum I et Q nam quater tantum, quanta ipsamet virtus ipsius S esse conspicitur.

Up to here we have come to know that the weight or the force of S which is half of that of I sustains the force of I and Q namely four times as much as the force of S is considered to be.50

Fig. 7.34: Second marginal note to the twenty-first chapter of Benedetti's book.

Fig. 7.34: Second marginal note to the twenty-first chapter of Benedetti's book.

Guidobaldo commented:

[…]dum hucusque verum est usque […]nem confuse immo sine [confusi]one totum hoc caput[…] est. Noluit [vero?] earum quae distincte in tractatu trochlea condimus repe[rire] sed ut aliquid novi af[figer]e videatur per ambages [ill]as communes species, et per communes conceptus ali[quid] [a]ttingit; tandem vero [aliquan]do aliqua vera profert [e]x nostro tractatu de [troc]hlea rescripsit. Quaequae hoc tractatum cap., neminem credo […] demonstrationemque trochlearum […]ere posse

[…] up to here everything is true and [not] confused. Even this entire chapter is without confusion. But he does not want to recover [anything] from that for which we have concisely provided the foundation in the treatise on the pulley but rather, in order to appear to add something new, he attacks something with the help of ambiguous ideas and common notions; nevertheless occasionally he advances something true [that] he has rewritten from our treatise on the compound pulley. Whatever […] is treated in this chapter, I believe that nobody can […] the proof of the compound pulleys

7.14 Letter to Pizzamano: simplifying the solution of a geometrical problem

The chapter is a letter entitled:

Qualiter circulus designari possit alios duos circulos propositos includens. Clariss. Petro Pizzamano.

How a circle can be designed so that it includes two other given circles. To the Most Brilliant Petrus Pizzamanus.51

Fig. 7.35: Drawing of a geometrical problem in Benedetti's book on which Guidobaldo commented.

Fig. 7.35: Drawing of a geometrical problem in Benedetti's book on which Guidobaldo commented.

This chapter belongs to the part of Benedetti's treatise in which he collected letters to illustrious personalities. The present letter is directed to Pietro Pizzamano who was, between 1559 and 1580, an official in Bergamo, Trevigi, and Mercanzia.52 It deals with a rather trivial geometrical problem to which Benedetti offered various solutions depending on the position of the two circles for which an encompassing circle is being searched (see figure 7.35). In his note at the bottom of the second page dealing with this problem Guidobaldo suggested one brief solution to the problem raised by Benedetti.53 While Benedetti's solution allows for a variable diameter of the surrounding circle, Guidobaldo's construction does not. Guidobaldo's solution works by simply dividing in half the line connecting the centers of the given circles and extending from one diameter to the other. He then concluded that a circle around this midpoint with this extension will touch the two given circles.

Guidobaldo's comment refers to the diagram at the bottom of page 263 and to Benedetti's text starting in the middle of the page:

Si vero distantia duorum propositorum circulorum tanta fuerit, quod secundi circuli nequeant se invicem tangere, vel secare, tunc alia via incedendum erit, quae talis est et generalis. Dividatur tota QB per aequalia in puncto Z circa quod signentur duo puncta ab ipso aequidistantia K et P. Distantia vero AK facta sit semidiameter esse unius circuli KX circa centrum A. Distantia autem OP semidiameter alterius circuli PX circa centrum O qui quidem circuli se invicem secent in puncto X a quo cum ductes fuerint XAD et XOF per centra dictorum circulorum, ipse erunt invicem aequales, eo quod cum BK aequalis sit QP igitur XD et QP erunt invicem aequales, sed FX aequalis est QP quare XF aequalis erit XD tunc si X centrum fuerit unius circuli, cuius semidiameter sit una dictarum, problema solutum erit.

Talis etiam solutio commoda erit ad inveniendum dictum circulum cuiusvis magnitudinis, dato tamen quod eius diameter, maior sit BZ cum in nostra potestate sit accipere puncta K et P proxima vel remota ab ipso Z ad libitum. Unde absque ulla divisione ipsius QB per medium, satis erit signare puncta K et P duabus distantiis mediantibus BK et QP invicem aequalibus, et etiam propositis.

But if the distance of the two given circles were such that the second circles cannot mutually touch or cut each other, then one has to proceed by another way which is as follows and which is general. Let the entire line QB be equally divided at the point Z around which two points K and P are marked which are equally distant from it. The distance AK shall be made the radius of one circle KX around the center A. But the distance OP shall be the radius of another circle PX around the center O which circles cut each other in the point X from which two lines XAD and XOF shall be drawn through the centers of the said circles, then these will be equal to each other, so that since BK is equal to QP therefore XD and QP are equal to each other, but FX is equal to QP so that XF is equal to XD, whence if X were the center of one circle whose diameter shall be that of one of those mentioned, the problem will be solved.

This solution will also be convenient to find the said circle of arbitrary magnitude, provided that its diameter is larger than BZ because it is in our power to assume the points K and P to be as close or distant from Z as we wish. Whence without any division of QB in half it will be sufficient to assign points K and P by two distances BK and QP equal among each other and also to the given ones.54

Guidobaldo commented:

In omnibus casibus, divisa BQ bifariam, quod quidem punctum fiat centrum, circulus descriptus per BQ transiens semper datos circulos in punctis BQ continget

In all cases, if BQ is divided in half, what makes, of course, that point to be the center, the circle described going through BQ will always touch the given circles in the points BQ

Fig. 7.36: Marginal note at the bottom of the first letter.

Fig. 7.36: Marginal note at the bottom of the first letter.

7.15 Letter to Mercato: rejecting an attempt to improve on Archimedes

The chapter is a letter entitled:

Considerationes nonnullae in Archimedem. Doctissimo atque Reverendo Domino Vincentio Mercato.

Some notes on Archimedes. To the most Learned and Reverend Lord Vincenzio Mercato.55

The chapter also belongs to that part of Benedetti's treatise in which he collected letters to illustrious personalities. In the present letter Benedetti dealt with

[…] duas Archimedis propositiones, quae in translatione Tartaleae sunt sub numeris .4. et .5. & in impressione Basileae sub numeris .6. et .7. […]

[…] two propositions of Archimedes that appear under the numbers 4 and 5 in Tartaglia's translation and under the numbers 6 and 7 in the Basel edition […].56

He aimed at improving Archimedes' arguments with which, as he also wrote,

[…] the mind cannot rest altogether satisfied.57

In his first argument he found a way of suggesting a derivation of the law of the lever by a procedure well known from Galileo's later treatise on mechanics, i.e. by redistributing weights with the help of a suspension mechanism that does not change the center of gravity.58 Imagine, to begin with, a balance suspended from its midpoint carrying two equal weights on its arms (see figure 7.37). In a first step Benedetti considered all parts of the weights evenly distributed along the entire length of the beam, with the center of gravity remaining in the middle. Then he imagined a construction by which the beam is suspended from a point right above the center of gravity by means of two lines (which Galileo later visualized as strings) which are positioned at unequal distances from the center of gravity. In a second step Benedetti considered the beam to be cut in such a way that each of the strings carries the broken parts of the balance from their centers of gravity. From the geometry of the situation and the initial assumption of the uniform distribution of weight the rest then follows, and is indeed left to the reader.

The second arguement deals with the steelyard and how its equilibrium is disturbed by moving the counterpoise. On this second argument Benedetti wrote:

Illa vero propositio, quam tibi dixi Archimedem tacuisse in huiusmodi materia est, quod si duo pondera aequilibrant ab extremis alicuius staterae, in certis praefixis distantiis a centro. Tunc dico si eorum uno manente alterum moveatur remotius ab ipso centro quod illud descendet, et si vicinius ipsi centro appensum fuerit ascendet.

The proposition about which I told you that Archimedes was silent deals with the subject of two weights in equilibrium at the ends of a steelyard at certain predetermined distances from the fulcrum. I say that, if one of these weights remains stationary and the other is moved farther from the fulcrum, that second weight will fall; while if that weight is appended nearer the fulcrum, it will rise.59

Fig. 7.37: Drawing for a proof of the law of the lever in the second letter of Benedetti's book with marginal notes of Guidobaldo.

Fig. 7.37: Drawing for a proof of the law of the lever in the second letter of Benedetti's book with marginal notes of Guidobaldo.

In his marginal comments Guidobaldo expressed little understanding for Benedetti's approach. Probably he was skeptical about Benedetti's pretension to improve on Archimedes. In his first comment Guidobaldo argued against Benedetti that it is impossible for both the midpoint of the beam of the balance and the point right above it to be centers of gravity of the weights under consideration. This was clearly a misunderstanding triggered by Benedetti's somewhat sloppy use of the word center for the point of suspension above the proper center of gravity.

More specifically, Guidobaldo's first comment refers to the passage at the top of page 381:

imagineris etiam OU quae sit parallela ipsi LK quae divisa sit in puncto I supra G. Hinc nulli dubium erit, cum G fuerit centrum totius ponderis appensi ipsi LK quod I similiter erit centrum cum directe locatum sit supra G hoc est in eadem directionis linea, quod quidem non indiget aliqua demonstratione, cum per se satis pateat.

Imagine also OU, parallel to LK and divided at point I above G. Thus no one can doubt, since G was the center of the whole weight suspended from LK, that I similarly will be the center, since it is situated directly above G, that is in the same line of direction. And this needs no demonstration, since it is quite clear by itself.60

Fig. 7.38: First marginal note to the second letter of Benedetti's book.

Fig. 7.38: First marginal note to the second letter of Benedetti's book.

Guidobaldo commented:

punctum I esse [cen]trum gravitatis. Deorsum pendet e[x] puncto G, quod est proprie ipso[rum] centrum gravita[tis] immo I non est centrum gravitat[is] ipsorum ponderum sed G.

[he claims] that the point I is the center of gravity [of the weights on the balance]. It hangs down from the point G which properly is their center of gravity therefore I is not the center of gravity of these weights but G.

In his second comment Guidobaldo caught another oversight by Benedetti. He criticized Benedetti for not correctly expressing the inverse proportion in the law of the lever, which is indeed the case. The second comment refers to the passage in the penultimate paragraph of the page (see figure 7.37):

Sit exempli gratia statera AU cuius centrum sit I et pondera U A appesa, se invicem habeant ut IU et IA se invicem habent.

Suppose, for example, that there is a steelyard AU, with fulcrum I and weights U and A appended, and suppose that they are to each other as IU to IA.61

Fig. 7.39: Second marginal note to the second letter of Benedetti's book.

Fig. 7.39: Second marginal note to the second letter of Benedetti's book.

Guidobaldo noted:

deest permutatio

the permutation is lacking

As mentioned above, Benedetti's idea of how to improve Archimedes' demonstration of the law of the lever was later taken up and elaborated by Galileo.

Footnotes

The transcriptions take into account the samples given by Anthony Grafton in the prospectus of the auction house (Catalogue 38 of Martayan Lan). For an analysis of the deletions, see the appendix.

Benedetti 1585, 141–142, pages 322324 in the present edition. Translation in Drake and Drabkin 1969, 166–167.

Benedetti 1585, 141, page 322 in the present edition. Translation in Drake and Drabkin 1969, 166.

Benedetti 1585, 142–143, pages 324326 in the present edition. Translation in Drake and Drabkin 1969, 168–169.

Benedetti 1585, 142, page 324 in the present edition. Translation modified from Drake and Drabkin 1969, 169.

Benedetti 1585, 143, page 326 in the present edition. Translation in Drake and Drabkin 1969, 169–170.

Benedetti 1585, 143, page 326 in the present edition. Translation in Drake and Drabkin 1969, 169.

Benedetti 1585, 143, page 326 in the present edition. Translation in Drake and Drabkin 1969, 169–170.

Benedetti 1585, 144–145, pages 328330 in the present edition. Translation modified from Drake and Drabkin 1969, 171–172.

Benedetti 1585, 145, page 330 in the present edition. Translation adapted from Drake and Drabkin 1969, 172.

Benedetti 1585, 144, page 328 in the present edition. Translation in Drake and Drabkin 1969, 171.

Benedetti 1585, 144, page 328 in the present edition. Translation in Drake and Drabkin 1969, 171.

Benedetti 1585, 145–146, pages 330332 in the present edition. Translation in Drake and Drabkin 1969, 172–174.

Benedetti 1585, 145–146, pages 330332 in the present edition. Translation in Drake and Drabkin 1969, 173.

Benedetti 1585, 148–149, pages 336338 in the present edition. Translation in Drake and Drabkin 1969, 174–176.

Benedetti 1585, 148, page 336 in the present edition. Translation in Drake and Drabkin 1969, 174–175.

Benedetti 1585, 148, page 336 in the present edition. Translation in Drake and Drabkin 1969, 176.

Benedetti 1585, 149–151, pages 338342 in the present edition. Translation modified from Drake and Drabkin 1969, 176–178.

Benedetti 1585, 150, page 340 in the present edition. Translation in Drake and Drabkin 1969, 177.

Benedetti 1585, 152, page 344 in the present edition. Translation in Drake and Drabkin 1969, 179–180.

Benedetti 1585, 152, page 344 in the present edition. Translation in Drake and Drabkin 1969, 179.

Benedetti 1585, 152, page 344 in the present edition. Translation in Drake and Drabkin 1969, 180.

Second marginal note in the tenth chapter.Benedetti 1585, 154, page 348 in the present edition. Translation modified from Second marginal note in the tenth chapter.Drake and Drabkin 1969, 182–183.

Benedetti 1585, 154, page 348 in the present edition. Translation in Drake and Drabkin 1969, 183.

Benedetti 1585, 155–159, pages 350358 in the present edition. Translation in Drake and Drabkin 1969, 184–187.

See the discussion in Büttner 2008.

Benedetti 1585, 158, page 356 in the present edition.

Benedetti 1585, 159–160, pages 358360 in the present edition. Translation in Drake and Drabkin 1969, 187–188.

Benedetti 1585, 159, page 358 in the present edition. Translation in Drake and Drabkin 1969, 187.

Benedetti 1585, 160, page 360 in the present edition. Translation in Drake and Drabkin 1969, 188.

Benedetti 1585, 162, page 364 in the present edition. Translation in Drake and Drabkin 1969, 190–191.

Benedetti 1585, 162, page 364 in the present edition. Translation in Drake and Drabkin 1969, 190.

See also the discussion in section 6.3.

Benedetti 1585, 162, page 364 in the present edition. Translation modified from Drake and Drabkin 1969, 191.

Benedetti 1585, 163–165, pages 366370 in the present edition. Translation in Drake and Drabkin 1969, 191–193.

Benedetti 1585, 163, page 366 in the present edition. Translation in Drake and Drabkin 1969, 192.

Benedetti 1585, 164, page 368 in the present edition.

Benedetti 1585, 262–264, pages 376380 in the present edition.

See the discussion in Bordiga 1985, 634.

Compare also Guidobaldo's discussion in his notebook DelMonte 1587, 148.

Benedetti 1585, 263, page 378 in the present edition.

Benedetti 1585, 380–396, pages 382414 in the present edition. Translation in Drake and Drabkin 1969, 235–237.

Benedetti 1585, 380, page 382 in the present edition. Translation in Drake and Drabkin 1969, 235–236.

Compare Favaro 1968, vol. 2, 161–163 and Galilei 1960a, 153–154 and the discussion in section 3.10.

Benedetti 1585, 381, page 384 in the present edition. Translation modified from Drake and Drabkin 1969, 236.

Benedetti 1585, 381, page 384 in the present edition. Translation in Drake and Drabkin 1969, 169.

Benedetti 1585, 381, page 384 in the present edition. Translation modified from Drake and Drabkin 1969, 236.