## Chapter structure

- 4.1 The prologue
- 4.2 The postulates
- 4.3 The first theorem on the proportion of descents and ascents of heavy bodies
- 4.4 The second theorem on the equilibrium position of a balance
- 4.5 The third theorem on the irrelevance of the lengths of the pendants
- 4.6 The fourth theorem on the decrease of positional heaviness
- 4.7 The fifth theorem on the descent of the longer arm
- 4.8 The sixth theorem on the bent lever
- 4.9 The seventh theorem on the freely swinging pendant
- 4.10 The eighth theorem on the law of the lever
- 4.11 The ninth theorem on the equal positional heaviness of bodies in different positions
- 4.12 The four theorems of the
*De canonio* - Footnotes

In the following we shall first briefly summarize the structure and the contents of the book annotated by Guidobaldo and then present his marginal notes in their context, with detailed explanations and quotations of the passages on which he chose to comment.

The *Liber Jordani de ponderibus* is based on fourteenth-century manu-

scripts edited by Petrus Apianus in 1533 at Nuremberg.^{1} It comprises a prologue, seven postulates, and thirteen theorems (see also sections 2 and 3.5). After each theorem, the Apianus edition presents a short commentary as well as a generally much more extended supplementary commentary, either by the editor himself or compiled from other manuscript sources, as Moody and Clagett contend.^{2}

## 4.1 The prologue

The prologue introduces the science of weights as being subject to both philosophy and geometry. The author then EOAemphasizes that the arm of a balance describes a circle. The effect of a weight in different positions of the arm of the balance is derived from the properties of motion along a circle, conceived in the manner of the Aristotelian *Mechanical Problems*. An arc is compared to its chord in order to assess its curvature. A longer arc of the same circle is thus considered to be more curved than a shorter one, and an arc of a given length is more curved in a smaller circle than in a larger one. The author furthermore assumed that the more curved a motion is, the more contrary it is to a straight line and the more violence it contains. But the more violence or more impediment the motion of a body acquires, the more its heaviness is diminished. This consideration constitutes the basis for the statement that a body becomes lighter the more the arm of a balance descends and ultimately for defining positional heaviness. After explaining how such considerations of contrariety of motions relate to a situation in which a heavy body is at rest, the author claimed to have prepared the postulates that are stated next. They are characterized as being in no need of proof and as constituting assumptions of the science of weights.

## 4.2 The postulates

The first two of the seven postulates (see page

## 4.3 The first theorem on the proportion of descents and ascents of heavy bodies

The first theorem specifies the second postulate. It states that the velocities of descent are in direct proportion to the weights of heavy bodies, while the contrary motions of descents and ascents are in inverse proportion to each other. This theorem lays the ground for a derivation of the law of the lever by starting from Aristotelian dynamics as it is then pursued in theorem 8. The direct proportionality between weight and velocity of descent is a common conclusion from Aristotelian dynamics (see section 3.4.1). The inverse proportionality characteristic of the law of the lever may then be derived from the contrariety of descent and ascent of the two arms of a lever. The first short comment makes this implication explicit, while the second, more detailed and technical comment discusses the conclusion in a scholastic style, refuting arguments of a possible adversary but also referring to propositions of Euclid and Archimedes.

## 4.4 The second theorem on the equilibrium position of a balance

The second theorem states that a balance with equal arms will not leave the horizontal position when two equal weights are attached. It also states that when the balance is brought into a different position it will return to the horizontal. The first commentary just refers to the fourth postulate for a justification of the latter statement, which is essentially to the definition of positional heaviness. This is hardly understandable without knowledge of the full argument in favor of this claim as it is familiar from other writings. The longer second commentary then provides the missing explanation showing how the fourth postulate is to be applied in order to arrive at the desired conclusions. The argument it provides is essentially identical to that of the *Elementa*, while the accompanying figure is somewhat different.

## 4.5 The third theorem on the irrelevance of the lengths of the pendants

The third theorem argues that the lengths of the supporting chords of the weights attached to the balance are inconsequential for the equilibrium. The first short comment sketches an indirect proof, arguing that if one body descends, the other side would have to be less heavy. This contradicts the premise that the weights on the two sides of the balance are equal. As a justification it refers, somewhat surprisingly, to the second postulate. The second longer comment essentially follows the reasoning of the *Elementa* and is based on the notion of *positional heaviness*. Remarkable is a concluding reference to the fact that the different distances of the suspension chords from the center of the world have been rightly neglected.

## 4.6 The fourth theorem on the decrease of positional heaviness

The fourth theorem claims that whenever a weight attached to the beam of a balance descends it becomes positionally lighter. The first commentary simply refers to the fourth postulate which states that a body becomes positionally heavier if its descent is less oblique. The second commentary then argues in more detail, again following the same logic as the *Elementa*, that when a weight is displaced from the horizontal position it will move through arcs that capture less of the vertical so that in fact the weight becomes, by the fourth postulate, less heavy positionally. This theorem is omitted from the *De ratione ponderis*.

## 4.7 The fifth theorem on the descent of the longer arm

The fifth theorem states that when the arms of a balance are unequal, but the weights attached are equal, the balance will descend on the side of the longer arm. The first commentary notes that the motion of the longer arm describes a larger circle and then refers to the third postulate which states that a body is heavier in descending if its movement toward the center is more direct. The extensive second commentary elaborates in great detail the mathematical relation between circle and arc and refers to Euclid, in particular to the edition of the *Elements* by Johannes Campanus,^{3} as well as to Ptolemy and Archimedes. The reference to Ptolemy's *Almagest*, with which Apianus was intimately familiar through his work on cosmography,^{4} suggests that Apianus, who around this time must have been working also on his famous sine tables,^{5} may have contributed his own thoughts to this commentary or even be its author, an issue which, however, remains controversial.

## 4.8 The sixth theorem on the bent lever

The sixth theorem is one of the problematic statements about the bent lever dropped from the *De ratione ponderis*. It states that when equal weights are suspended so that one weight is attached from the shorter arm in horizontal position, while the other weight is attached to the longer arm which is, however, bent so that its end is at the same distance from the vertical as is the shorter arm, then the weight on the longer arm will become positionally lighter. According to classical physics, as well as according to theorem 8 of the *De ratione ponderis* and Benedetti's rule, the two weights should, however, be in equilibrium. In the *Elementa* this erroneous conclusion is reached by arguing that the descent of the weight on the longer arm is more oblique than the descent of the weight on the shorter arm, comparing arcs that capture equal amounts of the vertical. The first comment to the theorem is again rather vague and hardly indicates an approach to demonstrating this conclusion. The more explicit second comment again follows the logic of the demonstration in the *Elementa*.

## 4.9 The seventh theorem on the freely swinging pendant

The seventh theorem states that when equal weights are suspended from equal arms, one by a freely moving chord of suspension and the other by a rigidly fixed rod at a right angle to the arm of the balance, the weight that is freely swinging will be positionally heavier. The theorem thus amounts to the surprising claim that the equilibrium of a balance depends on the mobility of its arms in such a way that the weight on the mobile arm has supposedly a greater effect than the weight on the arm that is fixed. What actually happens in this case according to classical physics is that such a balance in a horizontal position is in stable equilibrium, while a balance with two freely moving chords of suspension would be in indifferent equilibrium.

The first commentary is exceptionally long while still not being very helpful to the non-initiated. It begins as usual with an explication of the terms involved, followed by an indication of how the proof is to be carried out. The hint it gives, however, is limited to the enigmatic statement that the arm that can swing freely describes a greater circle in its descent. The demonstration found in the *Elementa* reduces this case to the bent lever considered in the preceding theorem. The arm of the balance with a fixed rod is mentally replaced by a bent lever along the hypothenuse of the triangle formed by the arm and the rod. The weight at the end of this bent lever has thus the same distance from the vertical line through the point of suspension as has the weight on the other arm of the balance. All that now remains is to compare the obliquity of the descents of these weights as it was considered in the previous theorem. The first comment proceeds by referring to the confusion that may arise when neglecting the difference between mobile and fixed arms, in particular when trying to establish the second theorem. As we have seen, the second theorem in fact claims that the balance always returns to the horizontal position, or in modern terms, that its equilibrium is stable. This, however, is certainly not the case when it has two freely moving chords of suspension. It is therefore remarkable that the first commentary explicitly states that theorem 7 was invented in the course of an experiment aimed at verifying theorem 2. It refers to the possibility that the claim of theorem 2 may seemingly be refuted by such an experiment if one considers a balance with freely moving chords. The second comment then provides the same argument as the proof of the *Elementa*.

## 4.10 The eighth theorem on the law of the lever

The eighth theorem states the law of the lever in terms of positional heaviness. The first commentary merely rephrases the claim and refers back to the first theorem. The proof of the *Elementa* is based on reducing the compensation of weights and lengths in a balance with unequal arms to a compensation of weights and heights to which they are lifted according to Aristotelian dynamics, using the preparation provided by theorem 1. The argument is in fact based on the statement inferred from theorem 1 that what suffices to lift a certain weight to a given height will also suffice to lift another weight to a different height if these weights and distances are inversely proportional to each other. The second commentary elaborates this idea in great detail, adding references to the relevant theorems of Euclid.

## 4.11 The ninth theorem on the equal positional heaviness of bodies in different positions

The ninth theorem states that two oblong bodies of equal weight and shape, one suspended in a vertical position, the other at its midpoint in a horizontal position, have the same positional heaviness. The first commentary first rephrases the theorem and then vaguely indicates that it can be proved by remarking that the semicircles described by these weights are equal. The proof of the *Elementa* decomposes the weight in horizontal position into two equal weights hung at equal distances from the midpoint of the original weight. It then shows that each of these weights balances half of the weight on the other side of the balance. The second commentary develops this idea and remarks in conclusion that this theorem constitutes, according to some, the end of Euclid's book on the balance.

## 4.12 The four theorems of the *De canonio*

The four remaining theorems are taken from the *Liber de canonio* (see also section 2). They deal with the weight of a material beam and its role for the equilibrium of a balance. The structure of the text remains the same. After the theorem a first commentary explains the terms, rephrases the theorem, and hints at a proof. The second commentary then provides technical details including a figure and typically referring to theorems of Euclid's *Elements*.

## Footnotes

The following description is based on Moody and Clagett 1960, 145–149. For a discussion of the extensive commentary and its manuscript basis, see also Moody and Clagett 1960, 293–305.

Johannes Campanus, 1220–1296.