The *Diversarum speculationum mathematicarum et physicarum liber* by Benedetti is a collection of six quite different treatises, as has been mentioned in the beginning (see section 1.2). Only one of these treatises entitled *De mechanicis* was in the focus of Guidobaldo del Monte's marginalia and will be presented here in some detail, summarizing and complementing what has been said in section 3.9.

This treatise on mechanics is divided into twenty-five chapters. There are some more references to mechanics in the letters that are also part of Benedetti's book. His discussion of the motion of fall through media and of hydraulic problems are not part of this treatise. The treatise starts with a brief preamble in which Benedetti claimed that he treats topics that have never been dealt with before or have not been sufficiently well explained.

## 6.1 The oblique position of the beam of a balance

Chapters 1 to 6 contain a systematic account of the foundation on which Benedetti built his mechanics. He presupposed the theory of Archimedes but also incorporated the concept of *positional heaviness*.

Chapter 1 clarifies qualitatively how the variable weight changes depending on the obliqueness of the beam of a balance. While a body attached to the end of the beam has a maximum weight if the beam is in a horizontal position, it vanishes when the beam is in a vertical position. Benedetti explained this behavior as a consequence of the different extent to which the attached weight rests on the center of the balance. If the position of the beam is close to the vertical, the weight of a body attached to the end of the beam is close to zero since it rests nearly completely on the center of the balance.

Chapter 2 clarifies the positional changing of the weight quantitatively. Benedetti related the balance with an oblique position of the beam to a bent lever with one horizontal and one oblique arm, thus providing the precondition for a generalization of his result. A generalization of this kind is indeed required if the lines of inclination of the bodies at the end of a balance are conceived as being directed to the center of the earth and hence no longer as being parallel to each other. Benedetti mentioned this possibility at the end of his chapter, but considered the angle between the two directions as being too small to be measured and thus need not be taken into account.

In chapter 3 Benedetti generalized from the downward inclination of a body attached to the beam of a balance to forces acting upon the body not vertically but making an acute or obtuse angle with the horizontal beam. Accordingly, he replaced the bodies at the end of the beam of a balance with two weights or two moving forces (*duo pondera, aut duae virtutes moventes*), as he formulated somewhat ambiguously. His derivation of their quantities was based on a reinterpretation of the horizontal distances between the center of the balance and the vertical projections of the bodies at the end of a beam in an oblique position. He interpreted these distances as perpendicular distances from the center of the balance to the lines of inclination, and was thus able to apply the result he achieved for vertically descending weights also to lines of inclination caused by forces that are not vertical.

In the following, Benedetti maintained that his arguments in chapters 1 to 3 clarify all the causes operating on balances and levers. To demonstrate this, he discussed in chapters 4 and 5 the validity of his results if applied to material balances and levers, taking into account that they have a beam with finite extension. This, however, does not imply that he calculated the influence of the weight of the beam itself. His discussion was rather restricted to a justification of his claim that the geometry of a rectangular beam does not require a modification of his propositions. In chapter 5 he treated the case of a lever whose fulcrum is at one of its ends.

Finally in chapter 6 Benedetti added the description of an instrument used in bakeries for treating the dough. He explained the function of the instrument by applying his proposition of chapter 3.

The systematic approach of Benedetti in this first part of his treatise is complemented by chapter 9 in which he justified the division of the scale of a steelyard into equal intervals.

## 6.2 Benedetti's criticism of Tartaglia and Jordanus

In chapters 7 and 8 Benedetti criticized theorems of his former teacher Tartaglia, in particular those Tartaglia adapted from Jordanus de Nemore. Both chapters deal exclusively with some propositions of Book VIII of Tartaglia's *Quesiti, et inventioni diverse*,^{1} which is concerned with the science of weights and is entitled, accordingly, *Sopra la scientia di pesi*. In those cases in which Tartaglia's propositions are adapted from Jordanus, Benedetti mentioned explicitly the corresponding proposition in the edition of Jordanus' *De ratione ponderis*, corrected and illustrated by Tartaglia, and published under the title *Iordani opusculum de ponderositate Nicolai Tartaleae studio correctum novisque figuris auctum*.^{2}

Chapter 7 starts with some brief critical remarks on Tartaglia's propositions 2 to 5. Tartaglia's proposition 2 essentially paraphrases and modifies the Aristotelian claim that the speed of moving bodies is proportional to the driving force. Following Jordanus, Tartaglia maintained that the velocities of descending heavy bodies of the same kind are proportional to their power (Italian: *potentia*) while in the case of ascending bodies their velocities are inversely proportional to their power. For bodies of the same kind their power is conceived here as proportional to their sizes, that is, to their weights. Descending bodies are thus simply falling bodies with velocities proportional to their weights, while in the case of ascending bodies their weight acts as a resistance. Tartaglia's proposition 3 generalizes proposition 2 for bodies with equal weights but unequal positional heaviness. His proposition 4 maintains that in the latter case the power of bodies attached to a balance is proportional to the distances from the center.

Benedetti's critical remarks are somewhat eclectic. He argued that Tartaglia, in his proposition 2, did not take into account the quantity of external resistance (*quanti momenti sint extrinsecae resistentiae*). With regard to Tartaglia's proposition 3 Benedetti pointed to its assumptions, namely that the bodies have to be homogenous and must have the same shape. He criticized that Tartaglia's proof does not actually require these assumptions but would be true also for heterogeneous bodies or for bodies with differing shapes. Concerning proposition 4 he criticized that Tartaglia did not prove what he claimed to prove. He should have rather followed Archimedes' proof of the law of the lever.

Benedetti's chapter 7 continues with a detailed discussion of the second part of Tartaglia's proposition 5 and the following two corollaries and is thus directly concerned with the equilibrium controversy. Tartaglia maintained in this proposition that a balance that is in equilibrium in a horizontal position will necessarily return to this horizontal position when moved into an oblique position. In a first corollary he claimed that the more the beam of a balance is brought into an oblique position, the more the bodies attached to it become positionally lighter. In a second corollary he claimed that while both bodies in this case become positionally lighter, the lifted body loses less of its positional heaviness than the body moving down. He concluded that the beam will return to a horizontal position. Benedetti questioned Tartaglia's approach by referring to the first three chapters of his own treatise, arguing in particular that Tartaglia's second corollary must be wrong. He discussed once more the beam of a balance in an oblique position, but now without the assumption that the lines of inclination of bodies attached to the beam of a balance are parallel. He rather considered the case that these lines are directed to the center of the world, showing, as we have discussed above in section 3.9, that not the lifted body but rather the body that is moved down loses less of its positional heaviness.

Benedetti continued in chapter 8 with critical comments on Tartaglia's propositions 6, 7, 8, and 14. Tartaglia's proposition 6 contains the proof of his fallacious claim that the lifted body of an oblique beam of a balance loses less of its positional heaviness than the body moving down, now modified by the further claim that the difference is smaller than any finite quantity. Tartaglia claimed:

[…] che la differenzia ch'è fra le gravità de questi dui corpi egli è impossibile a poterla dar, over trovar' fra due quantità inequali.

[…] that the differences between the heaviness of these two bodies is impossible to give or find between two unequal quantities.^{3}

Like Guidobaldo had done before him, but with different results, Benedetti criticized Tartaglia for not taking into account that the lines of inclination are not parallel.

Tartaglia's proposition 7 contains the simple statement that if the arms of a balance are unequal and bodies with equal weights are attached to the ends of the beam the balance will tilt on the side with the longer arm. Benedetti criticized that Tartaglia again did not take into account that the lines of inclination are not parallel and that in any case Tartaglia did not give the correct cause of the effect.

Tartaglia's proposition 8 formulates, following Jordanus, the law of the lever in terms of positional heaviness, stating that if the lengths of the parts of the beam of a balance with unequal arms are inversely proportional to the weights of the bodies attached to them, their positional heaviness will be equal. Benedetti criticized, just as Guidobaldo had done in his marginal notes to Jordanus, that this proposition is much better demonstrated by Archimedes. He added that therefore all the proofs of the propositions 9 to 13 are invalid.

Finally, Tartaglia's propositions 14 and 15 concern Jordanus' proof of the law of the inclined plane which, from a modern perspective, is essentially correct. Benedetti criticized, as we have also discussed in section 3.9, Tartaglia's argument by imputing to it an interpretation of the inclined plane as a balance, with the top of the plane being its center. His criticism based on the propositions of his chapters 1 to 3 thus completely missed the point of Tartaglia's argument.

## 6.3 Benedetti's criticism of Aristotle

Benedetti's treatise on mechanics continues mainly with critical notes on the Aristotelian *Mechanical Problems*^{4} which constituted a key reference for mechanical arguments at the time.^{5} His notes are as diverse as the Aristotelian *Mechanical Problems* themselves.

Before he embarked on this criticism, Benedetti dealt, in chapter 9, with the problem of why a steelyard carries a linear gradation.^{6} He took into account the weight of the beam and that of the scale by postulating the equilibrium of the balance when no extra weight is added. Then he added weights of one pound on both sides, arguing that, by common science (*scientia communis*),^{7} the balance stays in equilibrium if they are placed at equal distances from the fulcrum. He had thus found the mark on the beam that indicates a magnitude of one pound. He then successively placed further weights onto the scale, now arguing from the law of the lever that they must be compensated by distances proportional to their number. He thus avoided the problem of applying the law of the lever directly to a material steelyard, just as one does in practice when gauging such a balance.^{8}

In chapters 10 and 11, Benedetti started with critical remarks on Aristotle's first problem. Aristotle asked why larger balances are more accurate than smaller ones.^{9} Actually, this concrete physical question is not the focus of the extensive answer the author gave to this problem. He rather provided a long proof of the basic explanatory principle which plays a major role in the whole treatise (see section 3.4.1). At the end of the proof Aristotle argued that the same load will move faster on a larger balance thus making such balances more accurate.^{10}

The criticism Benedetti passed on Aristotle's argument has two parts. In chapter 10 Benedetti began by rejecting Aristotle's claim that the circumference of a circle combines concavity with convexity. He then argued against a specific part of Aristotle's proof of his principle which involves the superposition of motions. In this part Aristotle showed that:

Quandoquidem igitur in proportione fertur aliqua id, quod fertur, super rectam ferri necesse.

[…] whenever a body is moved in two directions in a fixed ratio it necessarily travels in a straight line.^{11}

He concluded:

Si autem in nulla fertur proportione secundum duas lationes nullo in tempore, rectam esse lationem est impossibile.

[…] if a body travels with two movements with no fixed ratio and in no fixed time, it would be impossible for it to travel in a straight line.^{12}

For the Aristotelian author this proposition served as a means to describe circular motion as a result of two movements with no fixed ratio. Benedetti, however, did not relate his criticism to this context. He argued only that Aristotle's inference concerning movements in two directions is not sufficient since a straight movement can result from two quite different motions, a criticism which does not really relate to the Aristotelian argument, other than showing that his entire attempt to derive the behavior of a balance from a principle of circular motion is misguided.

In the same vein Benedetti's criticism in chapter 11 then deals directly with Aristotle's answer to the question of why larger balances are more accurate than smaller ones. He argued that Aristotle's argument is not well founded since the greater accuracy has nothing to do with the motion of the beam of the balance but only with the geometrical constellation.^{13} To conclude he added a consideration of material balances, arguing according to his own principles that a weight on the larger balance will be positionally more effective.

Benedetti's chapter 12 concerns problems 2 and 3 of the Aristotelian *Mechanical Problems*.^{14} Problem 2 raises the question that forms the starting point of the equilibrium controversy:

Cur siquidem sursum fuerit spartum, quando deorsum lato pondere, quispiam id admouet, rursum ascendit libra: si autem deorsum constitutum fuerit, non ascendit, sed manet?

If the cord supporting a balance is fixed from above, when after the beam has inclined the weight is removed, the balance returns to its original position. If, however, it is supported from below, then it does not return to its original position. Why is this?^{15}

Aristotle implicitly assumed that the beam of the balance has a certain thickness and weight. It follows as a result of the geometry of the balance in an oblique position that if the beam is fixed from above, a greater part of the beam is on the lifted side of the perpendicular line across the suspension point (see figure 3.37). Consequently the beam will move back by itself into the horizontal position. The opposite is true for a beam fixed from below. In this case the greater part of the beam is on the lower side so that it cannot move back into a horizontal position by itself.

Benedetti criticized that in the first case it is not only the weight of the beam that causes it to return to the horizontal position but also the different distances of the weights in an oblique position from the vertical through the point where the beam is fixed. According to his theory of the dependency of the weight on the obliqueness of the beam, the weights must be different on both sides. Benedetti thus generalized Aristotle's argument to the case of a balance without a material beam carrying weight itself.

In the second case of a beam supported from below, he argued that Aristotle is completely mistaken. He rightly maintained that the beam will not remain in its oblique position, but that the lower part will further move down until the beam is below the point where it is fixed.

Problem 3 of the Aristotelian *Mechanical Problems*^{16} concerning an explanation of the effect of a lever is, to Benedetti, not worth the effort of a detailed criticism. He only briefly notes that Aristotle did not give the true cause which one will find in his own theory presented in chapters 4 and 5.^{17}

In the very short chapter 13, Benedetti criticized problem 6 of the Aristotelian *Mechanical Problems*:

Cur quando antenna sublimior fuerit, iisdem velis, et vento eodem celerius feruntur navigia?

Why is it that the higher the yard-arm, the faster the ship travels with the same sail and the same wind?^{18}

The Aristotelian answer provided in the *Mechanical Problems* is based on the interpretation of the yard-arm as a lever having its base where the yard-arm is fixed as the fulcrum. Benedetti maintained that this interpretation of the yard-arm as a lever:

[…] verum non est. Huiusmodi enim ratione navis tardius potius, quam velocius ferri deberet, quia quanto altius est velum, vi venti impulsum, tanto magis proram ipsius navis in aquam demergit.

[…] does not give the true explanation. For on this kind of explanation the ship would have to move more slowly rather than more swiftly. For the higher is the sail that is struck by the force of the wind, the more is the ship's prow submerged in the water.^{19}

Benedetti added one sentence with his own explanation according to which the ship with a higher sail moves more swiftly because the wind blows more strongly in the higher region.

Chapter 14 provides a long discussion of problem 8 of the Aristotelian *Mechanical Problems*. The question posed in this problem is why round and circular bodies are easiest to move. Three examples are mentioned and later discussed: the wheels of a carriage, the wheels of a pulley, and the potter's wheel. Benedetti claimed that Aristotle's answer to the question he posed is not sufficient. Nevertheless Benedetti himself argued essentially in a similar manner, only somewhat more extensively. Both of them argued that the circle, contrary to differently shaped bodies, touches a plane only at one point which can be considered as the fulcrum of a lever. But Benedetti added a further argument which is not given by Aristotle. He argued that a circle can be pulled along a plane without difficulty and resistance:

[…] quia huiusmodi centrum ab inferiori parte ad superiorem, nunquam mutabit situm respectu distantiae seu interualli, quae inter ipsum lineamqueADintercedit.

[…] because in such a case the center will never change its position by moving upward from below, i.e., will never change its position with respect to the distance or interval which lies beween it and lineAD.^{20}

At the end of the chapter Benedetti discussed the question of why a potter's wheel set into motion by an external force will continue to rotate for a time, but not forever. In his response he took into account the friction with the support of the wheel and with the surrounding air. But he also discussed reasons that are more deeply concerned with the nature of such motion. He claimed, in particular, that the rotational motion is not a *natural motion* of the wheel, evidently making reference to the Aristotelian distinction between natural and violent motions. He also claimed that a body moving by itself because an *impetus* has been impressed upon it by an external force has a natural tendency to move along a rectilinear path. This statement comes close to the principle of inertia of classical physics, although it actually deals with rectilinear motion as a forced motion and does not involve any assertion about its uniformity. Benedetti seems to suggest, in any case, that this natural tendency is in conflict with the forced rotational motion of the wheel, thus slowing it down, and the more so, the smaller the wheel and the more its parts are constrained to deviate from the rectilinear path.^{21}

In chapters 15 and 16 Benedetti dealt with issues of scale as they are brought up by the Aristotelian *Mechanical Problems*. In chapter 15, consisting merely of one short sentence, Benedetti referred to his own earlier treatment of Aristotle's question of why larger balances are more exact (erroneously citing chapter 10 instead of chapter 11 of his treatise) in order to deal with the ninth problem of the Aristotelian *Mechanical Problems* which reads:

Cur ea, quae per maiores circulos tolluntur et trahuntur, facilius et citius moveri contingit […]?

Why is it that we can move more easily and more quickly things raised and drawn by means of greater circles?^{22}

In chapter 16 he discussed the tenth problem of the Aristotelian *Mechanical Problems* which reads:

Cur facilius quando sine pondere est, movetur libra, quam cum pondus habet?

Why is a balance moved more easily when it is without a weight than when it has one?^{23}

In his detailed response to this problem – indeed much more detailed than the one found in the Aristotelian text – Benedetti compared two like balances with different sets of weights on their scales, one with two weights of one ounce, the other with two weights of one pound. He then added a half-ounce weight on one side of each balance and observed that the balance with the smaller weights moves more rapidly. He explained this effect by referring to the dynamical assumption that one always has to consider *the ratio of the moving force to the body moved*.

In chapter 17 Benedetti addressed the twelfth problem of the Aristotelian *Mechanical Problems* which reads:

Cur longius feruntur missilia funda, quam manu missa […]?

Why does a missile travel further from the sling than from the hand?^{24}

Benedetti's response is based on the concept of *impetus*, conceived as an intrinsic cause of motion originally acquired by the action of an external force that then gradually decreases after separation from the original mover. He argued that a greater impetus can be impressed by the sling due to the repeated revolutions which evidently lead to an accumulation of this intrinsic force. He observed that the impetus would lead, if not impeded by the sling or the hand, to a straight motion of the projectile along the tangent to the circle of its forced motion. He also noted – distancing himself from a claim made by Tartaglia – that the motion due to the impressed force can mingle with the projectile's natural motion downward, thus leading to a curved trajectory. It may well be the case that it was this claim that later induced Galileo and Guidobaldo to perform their experiment on projectile motion from which they drew the conclusion that such a mixture of motions indeed takes place.^{25}

In chapter 18 Benedetti considered problem 13 of the Aristotelian *Mechanical Problems* dealing with the question of why larger handles can be moved more easily around a spindle than smaller ones.^{26} In his short response Benedetti simply referred to the fourth and fifth chapters of his own treatise, stressing that everything depends on the lever and was evidently convinced that the Aristotelian reduction of such problems to properties of the circle is superfluous if not misguided.

In chapter 19 he handled in the same way problem 14 of the Aristotelian *Mechanical Problems* which reads:

Cur eiusdem magnitudinis lignum facilius genus frangitur, si quispiam aequi diductis [deductis] manibus extrema comprehendens fregerit, quam si iuxta genu?

Why is a piece of wood of equal size more easily broken over the knee, if one holds it at equal distance far away from the knee to break it, than if one holds it by the knee and quite close to it?^{27}

Again Benedetti just referred to the earlier chapters of his treatise.

In chapter 20 Benedetti reconsidered problem 17 of the Aristotelian *Mechanical Problems* which reads:

Cur a parvo existente cuneo magna scinduntur pondera, et corporum moles, validaque sit impressio?

Why are great weights and bodies of considerable size split by a small wedge, and why does it exert great pressure?^{28}

The answer is based on interpreting the wedge as two levers opposite to each other, their fulcra being placed at the entry points of the wedge into the wood. Benedetti disagreed with the identification of the two levers allowing the action of the wedge to be interpreted in terms of force, fulcrum, and resistance. He claimed that the fulcrum is actually placed just underneath the deepest point of the opening produced by the wedge entering a block of wood.

In chapter 21 Benedetti claimed to provide the true explanation of compound pulleys. He reduced a compound pulley to a chain of balances by appropriately identifying forces and fulcra, each wheel of the pulley corresponding to one balance.

In chapter 22 Benedetti discussed Aristotle's wheel, i.e. problem 24 of the Aristotelian *Mechanical Problems* which reads:

Dubitatur quam ob causam maior circulus aequalem minori circulo convolvitur lineam, quando circa idem centrum fuerint positi.

A difficulty arises as to how it is that a greater circle when it revolves traces out a path of the same length as a smaller circle, if the two are concentric.^{29}

While the author of the *Mechanical Problems* referred to dynamical reasons in explaining this apparent paradox, Benedetti resorted to a kinematic argument, a pointwise reconstruction of the trajectory of the motion of a point on the circumference, arguing that it results from a superposition of two motions. In the case in which the motion is controlled by the larger circle, a point on the circumference of the smaller circle traverses a path resulting from an *addition* of two motions. In the case in which the motion is controlled by the smaller circle, a point on the circumference of the larger circle traverses a path resulting from a *subtraction* of two motions.

Chapter 23 of Benedetti's treatise does not exist.^{30} In chapter 24 Benedetti discussed problem 30 of the Aristotelian *Mechanical Problems* which reads:

Cur surgentes omnes, femori crus ad acutum constituentes angulum, et thoraci similiter femur, surgunt?

Why is it that, when men stand up, they rise by making an acute angle between the lower leg and the thigh, and between the trunk and the thigh?^{31}

In his response Benedetti suggested that the reason for this behavior is to create an equilibrium of the body with regard to the line that serves as support underfoot.

In chapter 25 Benedetti addressed the last problem, problem 35 of the Aristotelian *Mechanical Problems* which reads:

Cur ea quae in vorticosis feruntur aquis, ad medium tandem aguntur omnia?

Why do objects which are travelling in eddying water all finish their movement in the middle?^{32}

Benedetti's answer simply referred to the fact that whirlpools are depressed in their middle without giving an explanation of this phenomenon. He could thus restrict himself to arguing that the motion of an object to the center of such a whirlpool is simply its natural downward motion. Remarkable is the final comment by Benedetti, concluding his criticism of Aristotle as well as his treatise on mechanics:

[…] a quo aliarum omnium quaestionum, quas ego omisi rationes sunt bene propositae.

But in the case of all those other problems that I have omitted, Aristotle's explanations are correct.^{33}

## Footnotes

Tartaglia 1546, 91r. Translation in Drake and Drabkin 1969, 130.

Aristotle 1980. See subsection 3.4.1.

See Rose and Drake 1971 and also the introduction to Nenci 2011a.

Benedetti 1585, 152, page 342 in the present edition, see also Drake and Drabkin 1969, 178.

In the sixteenth century the term *scientia communis* was used to designate knowledge common to all mathematical sciences, its core being the Euclidean theory of proportions. See Sepper 1996, 153–154.

See the discussion in Damerow et.al. 2002.

Aristoteles 1585, 507. Translation in Aristotle 1980, 337.

Aristoteles 1585, 508. Translation in Aristotle 1980, 339.

Benedetti 1585, 153, page 346 in the present edition; Drake and Drabkin 1969, 180–182.

Aristoteles 1585, 511. Translation in Aristotle 1980, 347–349.

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

Aristoteles 1585, 515. Translation in Aristotle 1980, 361.

Benedetti 1585, 155, page 350 in the present edition. Translation in Drake and Drabkin 1969, 183.

Benedetti 1585, 155, page 350 in the present edition. Translation in Drake and Drabkin 1969, 184.

For the historical context, see Büttner 2008.

Aristoteles 1585, 517. Translation in Aristotle 1980, 365.

Aristoteles 1585, 517. Translation in Aristotle 1980, 365.

Aristoteles 1585, 518. Translation in Aristotle 1980, 367.

See the discussion in Renn et.al. 2001.

Aristoteles 1585, 518. Translation in Aristotle 1980, 369.

Aristoteles 1585, 520. Translation in Aristotle 1980, 371.

Aristoteles 1585, 525. Translation in Aristotle 1980, 387.

In Drake and Drabkin 1969, 193 chapter 22 is erroneously numbered as chapter 23.

Aristoteles 1585, 532. Translation in Aristotle 1980, 403–405.

Aristoteles 1585, 533. Translation in Aristotle 1980, 409.

Benedetti 1585, 167, page 374 in the present edition. Translation in Drake and Drabkin 1969, 196.