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Hydrostatics on the fray: Tartaglia, Cardano and the recovering of sunken ships

Published online by Cambridge University Press:  08 April 2011

VIRGINIA IOMMI ECHEVERRÍA
VIRGINIA IOMMI ECHEVERRÍA
Affiliation:
Instituto de Historia, Pontificia Universidad Católica de Valparaíso, Paseo Valle 396, Viña del Mar, Chile. E-mail: virginia.iommi@ucv.cl.
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Abstract

In his 1551 Travagliata invenzione, the Italian mathematician Niccolò Tartaglia described a device for raising sunken ships. Despite his claim of originality, his contemporary Girolamo Cardano had described a similar method in his famous work De subtilitate, which was published a year before. A comparison between these methods reveals the uniqueness of Tartaglia's approach, for he combines an explicit defence of the horror vacui principle with an implicit negation of rarefaction. In this article I show the complexities of this conception and stress the importance of keeping the personal argument between both authors in mind when interpreting their descriptions of wreck-salvage operations.

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Type
Research Article
Information
The British Journal for the History of Science , Volume 44 , Issue 4 , December 2011, pp. 479 - 491
Copyright
Copyright © British Society for the History of Science 2011

In his Cronica of Italian mathematicians, Bernardino Baldi (1533–1617) devotes a brief but meaningful paragraph to the life and work of Niccolò Tartaglia (c.1500–1557). Baldi presents Tartaglia as a man of poor origins who was interested in geometry and arithmetic, and who had stood out in his own time for his ingenuity. Baldi stresses Tartaglia's relevance as a translator into Italian of Euclid's Elements, and he mentions without any specification Tartaglia's works on ballistics and on the fortification of cities, and the two volumes with speculations on pure mathematics. Baldi goes on to refer to Tartaglia's rivalry with Girolamo Cardano (1501–1576), and he concludes by criticizing Tartaglia's language, which made those who read his works laugh.Footnote 1 Tartaglia's style generated suspicions among scholars and was probably an impediment to the acceptance of his works in academic circles. Even today it constitutes a fundamental problem for their interpretation: it is complicated to follow Tartaglia's arguments, and sometimes it is difficult to determine whether Tartaglia had a clear idea of what he wanted to say.

The ambiguity of his thought has been taken as evidence of an intellectual process in which nature started to be analysed with new eyes. Historians of science, using his treatises on dynamics and mathematics, have looked into Tartaglia's ideas, stressing the complex and difficult path he took towards innovation.Footnote 2 This approach can be very useful for interpreting Tartaglia's conception of hydrostatics, an aspect of his work that has been mostly overlooked by historians.Footnote 3 For example, Baldi does not mention the two works Tartaglia wrote on the subject during the last years of his life, the Regola generale da sulevare con ragione e misura non solamente ogni affondata nave ma una torre solida di metallo (…) intitolata la travagliata inventione and the Ragionamenti sopra la travagliata inventione, both published in Venice in 1551. The first one is dedicated to Francesco Donato (1486–1553), a Venetian doge who had tried to raise a ship that had sunk near Malamocco. The work summarizes the Archimedean principles on the equilibrium of fluids. It also describes methods for raising a wreck and for descending safely into the depths of the sea. The edition of this treatise is accompanied by the Ragionamenti, in which Tartaglia gives an Italian translation of the first book of On Floating Bodies and describes the natural and physical reasons that support his invenzione.

Despite Tartaglia's claim for originality, Girolamo Cardano had given a brief exposition on the same subject a year earlier in his influential De subtilitate, a section of which describes how to raise a sunken ship. The long-lasting enmity between these two authors, which began after Cardano's publication of Tartaglia's solution to the cubic equation in 1545, seems to have had more than personal consequences: Tartaglia probably decided to study and explain more deeply a problem already analysed by his declared enemy.

Hydrostatics and sunken ships

Tartaglia began his treatise by describing the equivalence between the upward force exerted by water when an object is placed on it and the weight of the fluid displaced. He considered this physical principle as the starting point for proposing a practical method for raising a sunken object, and he recognized Archimedes as the most important theoretical referent. His device (see Figure 1) – composed of two empty ships united at the top by several logs – was placed above the wreck on the sea's surface. Each ship had a hole at the bottom with a beam that, when placed in the hole, impeded water from entering the ship. To recover the sunken ship, the beams would be removed and the ships on the surface would fill with water. Once the ships were full, the holes would be closed and the wreck, which would have risen to just beneath the surface, would be tied to the logs. Then, the water inside the ships would be drawn out with funnels. This would result in the final and complete ascension of the sunken ship.

Figure 1.

Tartaglia's device. From Niccolò Tartaglia, Travagliata inventione, Venice: Nicolo Bescarini, 1551. Source: Mathematica Italiana website, URL: mathematica.sns.it.

Tartaglia alluded very briefly to the sources that, apart from Archimedes, had inspired him to develop his device. In the tenth Dichiarazione of Book I of his Travagliata invenzione, Tartaglia mentions Vitruvius (first century BCE), Vegetius (fourth century CE) and Roberto Valturio (1405–1475) as authors who taught different ways of raising water. He refers to the variety of techniques proposed by authors in this tradition before putting forward his own.Footnote 4

Vegetius and Valturio alluded to the extraction of subterraneous waters but neither of them gave an explanation of the physical principles used. They concentrated mainly on the technical description of instruments.Footnote 5 However, in his De architectura, Vitruvius mentioned the case studied by Tartaglia explicitly. In Book Ten of this work, Vitruvius described in great detail the different kinds of machine involved in the movements of weights. It is not difficult to see how this book could have influenced Tartaglia: in the second chapter, Vitruvius described a machine made of two pieces of timber connected at the top with a brace for raising sunken objects. This machine was of a very similar construction to the one presented by Tartaglia in his Travagliata invenzione. He even might have followed Vitruvius’ hint that such a machine could be used for unloading ships.Footnote 6

If the connection to Vitruvius is evident, the relation to Vegetius and Valturio is not clear at all. Tartaglia probably decided to mention them because he was trying to connect his invenzione to well-known accounts of water clocks, pumps and diving appliances, and by these means to differentiate himself from the Renaissance tradition on wreck-salvage operations that had been developing since the fifteenth century. He omitted any reference to his direct predecessors on the subject and preferred to link his treatise to a parallel tradition that shared with him the practical approach on the mechanical motion of waters. Tartaglia probably wanted to stress the originality of his method and, by avoiding any allusion to the failed enterprise led by Leon Battista Alberti in 1446 to rescue the two ships of Caligula sunk at Lake Nemi, thought he might save himself from accusations of plagiarism.Footnote 7 Besides, he attributed to his invention a strategic and political meaning that was incompatible with a reference to the operation organized by the Roman Cardinal Prospero Colonna a century earlier. Tartaglia offered his device as a unique gift to the rulers of Venice for the enrichment of his city.

Although these intellectual and political reasons might explain Tartaglia's silence on the actions carried out at Nemi, they do not clarify another significant silence: in 1550, Girolamo Cardano published a method for raising sunken ships that undoubtedly challenged his enemy to propose a new and more practical device.

Cardano and the theory of elementary form

In the first book of the De subtilitate, Cardano presented a system for recovering a sunken ship based on the combination of weights. The analysis occured in the context of the motion of light bodies, and Cardano used it only as an example of his general explanation of motion. Cardano stated that, when ships are sunken in the sea, divers should bind them by ropes to vessels loaded with stones. Then the stones would be taken out and put in other vessels so that the emptied vessels drew the ship to them. This was possible, according to Cardano, because of the natural motion of air, which avoided being under water and thus would take the vessels, which had been almost sunken by the heaviness of the stones, up to the surface, pulling the tied ship with them. By the transference of stones from one set of vessels to another set located above, a ship could be brought to the sea's surface.Footnote 8

Cardano's presentation of the problem was fundamentally theoretical. He faced the question of establishing that the weight of the stones was equal to that of the water and to that of the ship that lay at the bottom of the sea. This equivalence must be combined with the motion of air towards its natural place. The solution might not seem very practical, for it involved a complicated and slow mechanism, and was probably inserted by Cardano only to strengthen his general theory of natural affection.

Cardano's theory of sublunary movement, which was based on his description of the elements, stands out as one of the most original produced at that time. He acknowledged the existence of a primary, underlying matter that did not change, and, instead of following Aristotle's conception of elementary motion (that is to say, towards natural places), he considered the movement of bodies to be determined by the rarefaction and condensation of matter.Footnote 9 According to Cardano, the elements could contract and expand up to a certain limit, depending on the circumstances, and attract or repulse any contiguous element. Using this principle, he completely transformed the traditional idea of natural motion. Cardano asserted that motions usually considered unnatural – of water going upwards, for example – were actually natural movements in the sublunary world. He established the predominance of the elementary form and the impossibility of the separation of the parts of matter, denying that the escape from a vacuum was a cause of motion. In fact, for Cardano, the conservation of form was the essential characteristic of elements, and it conditioned the usual motion towards natural places. Thus when a body could expand or rarefy more, it abandoned its natural movement to preserve its form, rising if heavy and descending if light.Footnote 10

Cardano based his explanation of the method for raising a wreck on the following single fact: air moved upwards with such force that it could pull even a sunken ship. His system was based on the traditional understanding of the natural movement of heavy things going downwards and of light ones going upwards. Although he omitted any reference to the notion of rarefaction as a consequence of the principle of the conservation of forms and natural affection, one can infer that when the stones were taken out of the vessels, the air contained in them expanded while the surrounding air entered and filled the emptied space. It was a combination of the traditionally accepted natural motion of light bodies with the theory of natural affection, which stressed the continuity of matter.

I have emphasized this last aspect mainly because Tartaglia presented a totally different theory for solving the same problem, and the distance between both solutions resided not only in the characteristics of the practical method used, but also in the natural principles that supported the respective methods.

Tartaglia's invenzione

Tartaglia based his invenzione on the following, simple principle: if two empty ships placed above a wreck were filled with water, the wreck would rise. He not only referred to the specific process by which water could be elevated, but also emphasized practical details, such as the speed with which the water inside the ships should be pumped out.Footnote 11

In addition, Tartaglia explained the natural mechanism that justified his method and gave a description of the behaviour of liquid media after being moved. He claimed that one who wants to separate a heavy body from the bottom of a deep stretch of water must move not only that body, but also all the water that lay above it. If the earthy bottom were liquid, he said, it would have to rise up to the space left by the body. However, because the bottom was immobile earth, it could not ascend to fill that vacant space. Instead, the surrounding water would accomplish that task. Tartaglia stated that the water that was first placed above the body descended to fill the vacant space left as the body was raised. Thus the water initially opposed the ascension and after a few seconds it occupied the bottom. This liquid medium was followed by more water that came to fill the spaces left by the body, helping the ascension. According to Tartaglia, that was the reason why it was much more difficult to separate a heavy sunken body from the bottom than it was to pull such a body to the surface once separated. This initial separation was even more difficult if the bottom was deeper, for in such a case there was more water above the body, and this water must be moved and raised with it.Footnote 12

The action that explained the entire process and turned an almost impossible task into a simple and practical mechanism was the continuous filling of vacant spaces left by the elevation of the sunken body. Tartaglia also alluded to the characteristics of the soil as something that could complicate the procedure. If the soil was stony, the sunken body would be surrounded and hugged by water except for the small part that touched the bottom. If swampy or sandy, the body would be embedded in the ground. In the first case, it was easier to separate the body from the bottom, for the water that filled the place left by the ascension of the body was already there, whereas in the second case it was necessary to bring far-off water to fill the whole swampy or sandy box, which took a lot of time. ‘Since nature does not allow void places to exist for a minute space of time’, Tartaglia said, ‘it is much more difficult to separate a body from a swampy or sandy bottom, than from a stony one’. As water needed time to fill the space, it was impossible to separate instantaneously all the parts of a body placed in such a box.Footnote 13

Elio Nenci has stressed that, even if some aspects of Tartaglia's and Cardano's expositions are quite similar, the Brescian author showed a deeper practical interest, for he focused on conditions that could make one case different from another.Footnote 14 In fact, Tartaglia's description of the method was less simplistic than Cardano's, for he included alternatives that related the efficiency of the method to the natural context. Tartaglia approached the difficulties caused in the salvage operation from the idea of the abhorrence of a vacuum. This medieval notion was well known at the time, but he quotes no authority for his position.Footnote 15 I believe that Tartaglia was probably inspired by a work that displayed a similar procedure based on the suction of water, namely the Pneumatica of Hero of Alexandria (first century CE).

In the description of the first device – the bent siphon – Hero explained the consequences of a vacuum's principle of attraction. According to Hero, it was possible to suck wine upwards with a siphon even though this was contrary to the nature of the liquid because, after the air inside the siphon had been extracted, a great pressure was exerted and the wine found no other place into which it could escape from it.Footnote 16 Although the situation to which Tartaglia refers was different – for he mentioned the consequences that the suction of water had on the lower strata of the sea – the logic of his answer was the same. Sucking off some of the water above the ship altered the state of natural equilibrium insofar as it altered the amount of water naturally above the ship. The result of this alteration was that the remaining water pushed the ship upwards and filled the empty spaces that were left behind as it did so.

Despite his description of the siphon, Hero defended the possibility of a vacuum. In the introduction to his Pneumatica, Hero explicitly denied the existence of a continuous void in nature.Footnote 17 But immediately after that, he proposed a subtle definition that functioned as the basis for his whole treatise and constituted its real peculiarity. Hero asserted that a vacuum was distributed in small measures through all bodies and used air as an example. According to Hero, the particles of air were in contact with each other, yet they did not fit together closely at every point. Rather, void spaces were left between the particles of air, as in the sand on the seashore. Thus when any force was applied to it, air was compressed and, contrary to its nature, fell into the vacant spaces by means of the pressure exerted on its particles. However, when the force was withdrawn, the air returned to its former position due to the elasticity of its particles.Footnote 18 Hero admitted that, when a continuous vacuum was produced by artifice, such a great pressure was exerted on the matter involved that it behaved contrary to its natural tendency. That is, despite his acceptance of a kind of void, Hero shared the fundamental idea of horror vacui, namely the direct relation between the unnatural motion of the elements and the resistance to an extended vacuum in nature. In other words, the experiments described in the Pneumatica could have been interpreted perfectly well in the light of the medieval theory of the abhorrence of a vacuum.Footnote 19 This ancient treatise might have helped Tartaglia to understand more clearly the way in which water could be used as a source of energy under determined circumstances. It also might have helped Tartaglia to focus on practical conditions, something that he could not find in Archimedes.

We do not have explicit evidence that Tartaglia was familiar with Hero's treatise. Antonio Favaro has analysed the official documents in which the Venetian Senate gave permission to Tartaglia for the publication of some mathematical works and has noticed an interesting fact. In a deliberation of December 1542, Tartaglia obtained an exclusive right to the publication in the city of his translations of and commentaries on Euclid, Archimedes and ‘Horone’.Footnote 20 As Favaro stresses, no work on ‘Horone’ has survived.Footnote 21 This right suggests that Tartaglia was familiar with some of Hero's mathematical work, but there is no proof of whether he was familiar with the Pneumatica. The only direct reference to Hero that we find in Tartaglia's work was a brief mention of Hero's method of inserting two mean proportionals in his Third Answer to Ludovico Ferrari.Footnote 22 The source in this case is indirect. Tartaglia clearly quoted, in exactly the same order, the exposition of different authors on the subject made by Eutocius in his Commentary to The sphere and cylinder. Footnote 23

Yet another element must be considered. The surviving registers of Bessarion's library in Venice for the years 1545 to 1559 record multiple loans of Greek codices. As Paul Lawrence Rose emphasizes, one of these borrowers is particularly noteworthy: the imperial ambassador to Venice, Diego Hurtado de Mendoza. The registers of 1545–1546 record the loan of the Pneumatica of Hero to Hurtado and, in view of Hurtado's friendship with Tartaglia, it is more than probable that Hurtado introduced this text to the mathematician.Footnote 24

It is also important to note that in the Pneumatica we find a reference that might be connected with Tartaglia's work. Using Archimedes’ treatise on floating bodies, Hero explained why divers were not crushed by the water above them, and he referred to the law according to which bodies of weight equal to that of a liquid would neither project above nor sink below the surface of that liquid when immersed in it.Footnote 25 Tartaglia himself described a winch, similar to the one used by Francesco de Marchi in Nemi in 1535, that could be submerged in deep waters.Footnote 26

The influence of Hero's explanation of the pressure exerted by a vacuum reveals a twofold aspect of Tartaglia's invenzione. On the one hand, by reviving horror vacui as the main cause of the raising of water, Tartaglia openly opposed the explanation offered by Cardano, and the Travagliata invenzione can be interpreted in light of their rivalry. On the other hand, the tacit reliance on the Pneumatica was limited to the description of devices and the practical application of the force produced by a vacuum, for Hero's theory of matter was totally omitted. This exclusion should be related to the absolute silence on the elasticity of bodies that characterizes Tartaglia's description of the phenomenon.

Rarefaction and the abhorrence of vacuum

The omission of the term ‘rarefaction’ stands out as a peculiar feature of Tartaglia's thought.Footnote 27 In the passages in which he referred to the detachment of the ship from the bottom of the sea, Tartaglia seems to follow the discussion of the separation of two surfaces in direct contact, which was a traditional subject of physical analysis. As Edward Grant has shown, most medieval authors agreed with Lucretius’ assumption that a momentary vacuum would be formed if two plane surfaces, with no material medium intervening, were separated in such a manner that they remained continuously parallel. ‘In rushing to fill the totally empty space between the now separated surfaces, the air would require some time, however small, to move from the outer perimeters to the innermost parts of the surfaces’, so it was concluded that separation could only occur gradually and successively, minute part after minute part.Footnote 28

Some authors did not limit their analyses to the classical case of two plane surfaces located in air that are pulled in opposite directions but mentioned also other situations. Roger Bacon and John of Jandun, for example, referred to the case in which a single surface came into contact with air or water, and they both stated that more air or water must arrive to fill the space produced by separation.Footnote 29

One can clearly see an influence of this medieval position in Tartaglia's invenzione. Apart from the evident resemblance, the mathematician's analysis was complemented by a fundamental role attributed to water, namely its double action as a filler and as a pusher, which makes his description of the phenomenon unique. This practical application of the theoretical explanation elaborated in the Middle Ages reveals Tartaglia's interest in silencing the role of rarefaction, for he not only reproduced a tradition that did not refer to the elasticity of matter in this particular case, but he also denied the possibility of expansion to a fluid soil (as found, for example, in a swamp) and explained the movement of masses of water without referring to the expansion of the element. The importance of this omission can be understood by considering the relevance that rarefaction had in sixteenth-century discussions on the existence of void as a basic characteristic of matter. According to these discussions, rarefaction confirmed either the capacity of matter to increase in volume or the presence of void pores inside it.Footnote 30 In those years, the classical approaches found a renewed relevance. The Aristotelian position, which considered the contraction and expansion of matter as a qualitative rather than a quantitative change, was challenged by the atomist approach, which denied the possibility of the rarefaction of matter and interpreted it as a proof of the existence of void.

As Grant has stated, the medieval tradition that criticized the approach of Bacon and John of Jandun used rarefaction as its central argument. This tradition relied on the claim by Aristotle in De anima II, 11 423a, that surfaces in a medium could not come into direct contact since a portion of the medium itself would always intervene. Among the authors who defended this position was the English logician Walter Burley, who stated that the intervening film of matter would expand and fill the widening space.Footnote 31

This second explanation seems to have been more popular among Italian authors writing in the middle of the sixteenth century than the one followed by Tartaglia. The significance of Tartaglia's omission of the notion of rarefaction can be seen by comparing his account of the separation of two adjacent bodies (ship and soil) with others given at the time. Alessandro Piccolomini faced the same problem in his 1551 La prima parte della filosofia naturale.Footnote 32 Piccolomini stated that, when two flat and clean bodies were separated, a void was necessarily produced between both of them before air could flow in and fill the space. But, he said, due to the imperfection of matter, it was impossible to find bodies made of natural substances that had surfaces so flat that they could touch so completely in water or air, ‘and that a particle of air and water would not be enclosed, which in the separation of the said bodies, would rarefy and consequently not make necessary the presence of vacuum’.Footnote 33

Piccolomini strictly followed the Aristotelian assumption and reproduced the allusion to rarefaction made by Burley. So did Cardano, who referred to this question in the second edition of his De subtilitate, which was published in 1554. There, he included a passage on the separation of flat objects in which he claimed that air filled the separation that otherwise would constitute a vacuum. According to Cardano, since flat things were exactly united, less air was included between them, so it was more difficult to separate them. ‘The reason’, wrote Cardano, was ‘that it is the more necessary that air separates from its proper form the more it rarefies’.Footnote 34

This explanation reveals the main difference between Cardano and Tartaglia. While the latter held that for raising a wreck it was necessary that the surrounding water converged at the bottom of the ship to avoid a vacuum, Cardano held that when two surfaces were united, there was a small separation only because the matter between them was reduced and had a limited rarefaction. So, although they agreed on the necessity of more matter to complete the separation, their basic physical assumptions differed radically. It is likely that Tartaglia defended the concept of horror vacui to explain this process as a way of stressing the impossibility of eliminating this principle and thereby criticizing Cardano's theory. And maybe the Milanese philosopher added this passage to express clearly his idea concerning this phenomenon and to pose a diverse explanation based on the strictly Aristotelian tradition of Burley.

I believe Tartaglia's preference for the non-Aristotelian tradition in this particular passage of his treatise was a consequence of his notion of matter, which differentiated him from his contemporaries. Tartaglia conceived the medium as a continuous body in which motion was produced and that had a defined and unalterable extension. Unlike his medieval predecessors – who did not deny rarefaction as a property of matter, but only omitted it in the interpretation of a specific phenomenon – Tartaglia never mentioned the elasticity of media.Footnote 35 For him, not only was there no expandable medium intervening between the wreck and the soil, but also the water located at its sides was inextensible.

Footnotes

n. 23

References

1 Bernardino Baldi, Le vite de'matematici, Milan: Francoangeli, 1998. This passage of the Cronica is quoted in p. 51 n. 61.

2 See Duhem, Pierre, Les origines de la statique, Paris: Hermann, 1905Google Scholar; idem, Etudes sur Léonard de Vinci, Paris: F. De Nobele, 1913; Alexandre Koyré, ‘La dynamique de Nicolo Tartaglia’, in idem, Etudes d'histoire de la pensée scientifique, Paris: Presses universitaires de France, 1966, pp. 101–121; Costabel, Pierre, ‘Vers une mécanique nouvelle’, in Sciences de la Renaissance. VIIIe Congrès International de Tours, Paris: Vrin, 1973, pp. 127–142Google Scholar; idem, ‘Observations et théories du mouvement au XVIème siècle’, Actes des Journées Galilée, Cahier du seminaire d'historie des sciences et d'epistemologie (1980) 7–8, pp. 1–16; Henninger-Voss, Mary J., ‘How the New Science of cannons shook up the Aristotelian cosmos’, Journal of the History of Ideas (2000) 63, pp. 371–397CrossRefGoogle Scholar; Strong, Edward W., Procedures and Metaphysics: A Study in the Philosophy of Mathematical–Physical Science in the Sixteenth and Seventeenth Centuries, Hildesheim: Olms, 1966Google Scholar; De Pace, Anna, Le matematiche e il mondo. Ricerche su un dibattito in Italia nella seconda metà del Cinquecento, Milan: Francoangeli, 1993Google Scholar.

3 His ideas on hydrostatics have been taken into account in broad analyses of the Renaissance Archimedean revival. See Keller, Alex, ‘Archimedean hydrostatic theorems and salvage operations in 16th-century Venice’, Technology and Culture (1971) 12, pp. 602–617CrossRefGoogle Scholar; and Meli, Domenico Bertolini, ‘Guidobaldo del Monte and the Archimedean Revival’, Nuncius (1992) 7, pp. 3–34CrossRefGoogle Scholar.

4 Tartaglia, Niccolò, Travagliata inventione, Venice: Nicolo Bescarini, 1551Google Scholar. Book I, Dichiarazione 10. ‘Anchora, che Vitruvio, Vegetio, e Valturio ne insegnino varii, e diversi modi per condure acqua in alto, delli quali molti se ne potriano accomodare in questa nostra inventione, per commodità de impire, et vodare tutte le sorte de legni, over vasi per avanti detti, de lo quali anchora molti ne sono notissimi, et famigliarissimi, cioè con trombe, con rote, con mantici, con istrumenti incavati a vida, e molti altri.’

5 See Flavius Vegetius Renatus, Epitoma rei militaris, Book IV, X; and Roberto Valturio, De re militari, Verona: Johannes Nicolai de Verona, 1472.

6 Vitruvius, On Architecture (ed. and tr. Frank Granger), Cambridge, MA: Harvard University Press, 2002, p. 145. As Keller has stated, apart from the references mentioned by Tartaglia, an Italian tradition on this specific problem can also be identified, as both Mariano Taccola and Leonardo da Vinci had proposed similar devices. The former was the first to suggest the possibility of raising a chest from a wreck by attaching it to an anchor lowered from a dinghy full of stones that were thrown overboard and thus allowed the rising of the anchor and the chest. The latter had a note on the method for transporting a great weight over water by flooding a barge. See Keller, op. cit. (3).

7 Alberti mentioned this operation in his De re aedificatoria (Book V, Chapter 12), and he supposedly explained it in more detail in his now lost Navis. This treatise was well known in the Renaissance, and Leonardo da Vinci possessed a copy of it. See The Notebooks of Leonardo da Vinci (ed. Jean Paul Richter), 2 vols., New York: Dover, 1970, vol. 2, 1113, pp. 273–274. Tartaglia had already been accused of plagiarism by Ludovico Ferrari in 1547, when the latter noticed the similarity between the mathematician's exposition of the medieval science of weights in his Quesiti et inventione diverse (Venice, 1546) and Jordanus Nemorarius's De ponderibus.

8 Girolamo Cardano, De subtilitate (ed. E. Nenci), Milan: Francoangeli, 2004, pp. 77–78. ‘At motus levis hoc unum subiiciatur exemplum. Cum naves freto merguntur, quas eruere consilium est, cimbae [‘cymbae’ in later editions] onustae saxis per funes alligantur navigio ab urinatoribus, sic ut funes quantum fieri potest tendantur, inde totidem cimbis vacuis lapides ex prioribus detracti excipiuntur; quo fit ut allevatae cimbae navigium paululum ex profundo secum trahant. Nam aer cimbas, quae pondere lapidum ferme mergebantur, cum aquae subesse nolit, in superficiem aquae attollit; unde navigium ferme pro cimbae altitudine superius trahitur. Trahatur igitur ex A in B, tunc cimbae quae plenae sunt lapidibus illi annectantur funibus, transfusisque lapidibus navigium trahetur in C. Rursus priores cimbae, in quas lapides transfudisti, nectuntur tensis funibus navigio in C existenti, trahentque deductis lapidibus ipsum in D, atque perpetua transmutatione ad aquae superficiem tandem deducetur.’

9 Cardano, op. cit. (8), pp. 61 and 64.

10 Raffaelo Caverni considered this negation of the principle of horror vacui as a revolutionary new doctrine that determined subsequent scientific discoveries. Caverni, Raffaelo, Storia del metodo sperimentale in Italia, 6 vols., Florence: G. Civelli Editore, 1891–1900, vol. 3, p. 435Google Scholar. However, this assertion is exaggerated. Pierre Duhem traced back to Roger Bacon the first positive conception of natural affection as an alternative to the abhorrence of a vacuum, whilst Lynn Thorndike argued that Adelard of Bath was in fact the first to formulate a theory of universal continuity in his Natural Questions. See Duhem, Pierre, Le système du monde: Histoire des doctrines cosmologiques de Platon à Copernic, 10 vols., Paris: Hermann, 1954–1965, vol. 8, pp. 143 and 146–147Google Scholar; Thorndike, Lynn, A History of Magic and Experimental Science, 8 vols., New York: Columbia University Press, 1923–1958, vol. 2, pp. 37–39Google Scholar; and Grant, Edward, Much Ado about Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution, Cambridge: Cambridge University Press, 1981, pp. 67–69CrossRefGoogle Scholar.

11 Tartaglia, op. cit. (4), Book I, Dichiarazione 10.

12 Tartaglia, Niccolò, Ragionamenti, Venice: Nicolò Bescarini, 1551Google Scholar. Dialogue with Richard Wentworth, Second Reasoning. ‘Eglie così chiaro che a voler levare, et separare un corpo grave, dal fondo di qualche profonda acqua, non solamente eglie necessario a movere quello ma anchora a sulevare primamente tutta quella acqua che se gli riposa sopra. Et quando che il terreo fondo fusse acqua sarà necessario ad ellevarsi in parte de sotto in suso per venir a impire quel luoco che andasse lassando de mano in mano quel tal corpo, che si elevassi, ma per esser quel tal fondo immobile terra, la qual non potendo ascendere alla repletione di quel tal luoco vacuo eglie necesario che la maggior parte di quella acqua lo circonda dalle bande scorra a far questo tal officio, cioè a riempire quel tal luoco, et nel discendere tal acqua se tira anchora a drio gran parte di quella che doveria seguire a quella che sopra sta a quel corpo il che causa maggior difficultà in quella al moversi di sotto in suso: la qual combustione (per esser la detta acqua di natura grave) non poco, ovia all'ascenso del detto corpo in quel principio, ma immediate che si venghi a separare alquanto dal detto fondo, tanto è la furia de l'acqua che vi concorre, per impir quel luoco, dal qual si parte qual tal corpo (per non lasciarlo vacuo) che immediate comincia a dar aiutto è favore, alla elevatione di quello et continuando tal attratione parte de sotto giacente acqua, lo va seguitando con altra che vi concore insieme per impire, continuamente li detti vacui luochi che di continuo va lassando, il corpo ellevato, et quella acqua che sopra sta a quel tal corpo (per esser già commossa, et verso la parte superior) non da tanto impedimento all'assendimento dil detto corpo, anci per la commozione già fatta in quella, vien a esser fatta molto consentiente all'assendimento di quello et questa è la causa che eglie molto maggiore difficultà a separar dal fondo ogni cosa grave affondata, di quello che sarà poi che sia separata a tirarla in pelo d'acqua, e questa difficultà tanto più sarà maggiore quanto, che in fondo più profondo sarà, perché l'acqua che vi ripossa sopra (qual se ha da comovere e sulevar in parte insieme con lui) è di maggior quantità.’

13 Tartaglia, op. cit. (12), Dialogue with Richard Wentworth, Second Reasoning. ‘Hor perchè sia mo tanta, e tanta difficultà a separarlo da un fondo pantanoso, over arenoso di quello che sia da uno sassoso, la causa è questa, che in un fondo sassoso tutto il detto affondato corpo è abbrazato e circondato dal'acqua eccettuando quella poca parte che tocca il detto fondo sassoso, la qual parte ancora quanto, che è più accuta, cioè che tocca mancho del detto fondo tanto è piu facile a separarlo da quello, perchè l'acqua che ha da impire quell luoco, che lassarà il detto corpo nella sua assensione, è ivi presente, cioè che non ha da venire da loco molto lontano, e però il detto corpo non havendo tanta difficultà a tirare da longinque parti l'acqua che ha da impire quell suo loco, che ha da lassare, quantiche haveria a doverla tirare da longique parti, come che gli occorrerai quando, che fusse in gran parte sepolto nel pantano, over sabbia, nella qual positione, gli bisognaria tirare la detta acqua dalla suprema parte di quella sua cassa pantanosa, over arenosa per fin nella infima parte di quella, et perché tal acqua non può così immediatamente, over in un instante discorrere in tal parte infima, ma solamente in tempo. Et la natura non permette che un loco possi restar vacuo per alcun minimo spacio di tempo, e perciò è cosa molto, e molto più difficoltosa a separar un corpo grave da un fondo pantanoso, over arenoso, di quello sarà in un fondo sassoso … Dico che tal sarà necessario che quel tal loco restasse per un poco di tempo vacuo, cioè senza corpo perchè l'acqua (per esser corpo grave) non potrà scorrere et impire tutto quel vacuo in un instante, anci vi scorrera solamente in tempo. Et perchè la natura non permette che alcun loco possi restar vacuo per alcuna minima parte di tempo (come di sopra è detto) seguita adunque; esser impossibile a separare un tal corpo in un istante da tutte le parti di tal cassa pantanosa, over arenosa, et tanto maggior sarà tal difficultà, quanto che tal sua cassa sia fata universalmente contigua con el detto corpo, e indurata dintorno a quello.’

14 Cardano, op. cit. (8), pp. 78–79 n. 23. Nenci has stressed Cardano's and Tartaglia's common interest on the subject, but has not provided an explanation of their differences on the basis of their conception of matter.

15 See Grant, Edward, ‘Medieval explanations of the dictum that nature abhors a vacuum’, Traditio (1973) 29, pp. 327–355CrossRefGoogle Scholar.

16 Hero of Alexandria, Opera, 5 vols., Stuttgart: Teubner, 1976, vol. 1, pp. 28 and 30.

17 Hero of Alexandria, op. cit. (16), p. 4.

18 Hero of Alexandria, op. cit. (16), pp. 6–8.

19 Giorgio Valla, in his De expetendis et fugiendis rebus, repeated some of the experiments by Hero, but omitted any reference to his theory of matter. Francesco Maurolico, who probably used this work for his Ex Heronis et aliorum spiritalibus (Biblioteca Nazionale Centrale di Roma, San Pant.115/32, c.43r–44v; 46r–47v), described the mechanism of Pneumatica's first device as a consequence of the impossibility of a vacuum in nature. This shows that the experiments in themselves were not necessarily interpreted as the effect of porous matter.

20 ‘Viene in appresso la deliberazione sotto il dì 11 di dicembre 1542 con la qual vien'concesso al fidel nostro Nicolo Tartalea che alcuno non possa senza permissione sua stampar ne far ristampar in questa città ne in alcun altro luogo nostro Euclide et Horone philosopho per lui tradotti e commentati, et Archimede.’ Quoted in Favaro, Antonio, ‘Di Niccolo Tartaglia e della stampa di alcune delle sue opere con particolare riguardo alla Travagliata Invenzione’, Isis (1913) 1, pp. 329–340CrossRefGoogle Scholar, especially p. 330.

21 It might appear possible that he referred to Hero's De mensuris as it was usually part of the medieval Archimedean corpus, but since Tartaglia used for his edition MS Madrid, Biblioteca Nacional 9119, which does not contain this text, this alternative seems improbable. For a description of this manuscript see Clagett, Marshall, Archimedes in the Middle Ages, 10 vols., Madison and Philadelphia: The American Philosophical Society, 1964–1984, vol. 2, pp. 60–71Google Scholar.

22 ‘Ma saranno ineguale tal problema si potera con industria risolvere con modi phisicali cioè per quel modo dato da Platone per ritrovar due linee medie proporzionale fra due linee date, over per quel modo trovato da Herone’. Niccolò Tartaglia, Cartelli di sfida matematica, facsimile reproduction of the first editions published in 1547 and 1548, Brescia: Ateneo, 1974, p. 80.

23 Eutocius mentions the solutions by Plato, Heron, Philon, Apollonius, Diocles, Pappus, Sporus, Meneachmus, Archytas, Eratosthenes and Nicomedes. See Archimedes, Opera Omnia (ed. I.L. Heiberg), 4 vols., Stuttgart: Teubner, vol. 3, pp. 57–99. Eutocius’ commentary was part of the translation made by William of Moerbeke, a work on which Tartaglia based his 1532 Italian edition of Archimedes’ texts. Since the answer to Ferrari dates from 1547, it is also possible that Tartaglia used, as Marshall Clagget says, the Archimedean Basel edition of 1544, which also contained this work by Eutocius. See Clagett, op. cit. (21), vol. 3, pp. 547 and 549–551. He might have also taken the reference from Valla's De expetendis et fugiendis rebus, which contained a Latin translation of this section of Eutocius’ commentary.

24 Rose, Paul Lawrence, The Italian Renaissance of Mathematics, Geneva: Librairie Droz, 1975, p. 46Google Scholar. See also Castellani, Carlo, ‘Il prestito dei codici manoscritti della biblioteca di San Marco in Venezia ne'suoi primi tempi a le conseguenti perdite de'codici stessi’, Atti del Reale Istituto Veneto di Scienza, Lettere ed Arti (1896–1897) 7(8), pp. 311–377Google Scholar. Mendoza made a copy of the manuscript in 1542. The other works contained in the codex are quite revealing of Tartaglia's mathematical influence: ‘Manuel Moscopulus de Inuentione quadranguli. Maximi Planudis Calculatio secundum Indos. Nicolai Artabasdi Traditio calculatoriae disciplinae. Eiusdem calculati … Isaac Argyri ratio dimetrendi agros …’. Hobson, Anthony, Renaissance Book Collecting: Jean Grolier and Diego Hurtado de Mendoza, Their Books and Bindings, Cambridge: Cambridge University Press, 1999, p. 236Google Scholar, item 71.

25 Hero of Alexandria, op. cit. (16), p. 24.

26 Francesco de Marchi explored the wrecks in Lake Nemi using a diving bell, an enterprise described in his Della architettura militare, Brescia: Dall'Oglio, 1599.

27 One cannot argue that he was not familiar with this concept for he had read Aristotle's Physics (Tartaglia quotes it in his Nova scientia), in which it is repeatedly used. See Book IV, 9, 217a 12, where Aristotle explains that a given matter can change into a smaller or larger mass. See also Book VIII, 7, 260b 5–14; Book IV, 5, 212 b3. Aristotle also refers to rare matter in Book IV, 9, 217b 10, 12.

28 Grant, op. cit. (10), pp. 87–88. See Lucretius, De rerum natura, I, 384–390. See also Grant, op. cit. (15). Some authors that defended this thesis were Roger Bacon, Pseudo-Siger of Brabant, John of Jandun and the fathers of Coimbra. This problem was also solved using a theological argument. For example, Henry of Ghent (1217–1293) argued that God might create a void in this circumstance; see Alexandre Koyré, ‘Le vide et l'espace infini au XIVe siècle’, in idem, Etudes d'histoire de la pensée philosophique, Paris: Gallimard, 1971, pp. 37–92.

29 Grant, op. cit. (15).

30 Schmitt, Charles B., ‘Experimental evidence for and against void: the sixteenth-century arguments’, Isis (1967) 58, pp. 352–366CrossRefGoogle Scholar.

31 Grant, op. cit. (15).

32 Although Tartaglia acquired a copy of this work at some point, we cannot assume that he was responding to Piccolomini with his invenzione. Tonni-Bozza, ‘Frammenti di nuove ricerche intorno a Niccolò Tartaglia’, in Atti del Congresso Internazionale di Scienze Storiche (Roma 1–9 Aprile 1903), vol. 12, Rome: Tipografia della R. Accademia dei Lincei, 1904, pp. 293–307. This work stands out as one of the first comprehensive works on natural philosophy written in Italian. In Piccolomini's dedication to Pope Julius III, he refers to those few who had written in the vernacular on particular philosophical problems. Rufus Suter, ‘The scientific work of Alessandro Piccolomini’, Isis (1969) 60, pp. 210–222. One of these authors was probably Tartaglia.

33 Piccolomini, Alessandro, La prima parte della filosofia naturale, Venice: Vincenzo Valgrisi, 1551, pp. 225–226Google Scholar: ‘si vede sensatamente che posti insieme due corpi lunghi, e ben piani e politi, in modoche con la pianezza de le lor faccie si tochino perfettamente, subito poi che noi gli separiamo, sarà forza che nel mezi tra i lor piani, si trovi qualche voto d'ogni corpo, prima che l'aer possa trascorrere e riempire per ogni parte, non si potendo dar movimento subito et senza tempo, secondo che al suo luogo dichiararemo, a questi tali, che così dubitassero, rispoderei che per la imperfettione de la materia non si possano tra le sostanze naturali trovar corpi così piani e politi be le superficie loro, che così unitamente o in acqua, o in aere toccar si possino, che qualche particella d'aere o d'acqua non s'interchiuda, la qual particella nel separarsi de i detti corpi, vien a rarefarsi e conseguentemente non porge necessità che vi si debbi trovare il voto.’

34 Cardano, op. cit. (8), p. 74: ‘quod plana quae non fatiscunt, attamen disiunguntur. At hoc fieri nequiret, nisi admisso vacuo. Inter plana igitur quaecunque dum clauduntur, aer intercipitur, qui vetat plana disiungi quoad licet. Sed cum, ut dixi, pro sui robore solum impediat, maiore nixu vincitur. Solum illud obiicies, quod a minore aere minus haec disiunctio impediretur, at magis tamen impeditur. Ut enim magis plana ad unguem coierint, eo minus aeris intercipitur, et tamen eo difficilius divelluntur. Sed causa est, quia magis necesse est aerem illum a propria forma recedere, quo magis rarescit.’ On Cardano's three different editions of De subtilitate see Paola Pirzio, ‘Note sulle tre redazioni del De subtilitate di Girolamo Cardano’, in Marialuisa Baldi and Guido Canziani (eds.), Girolamo Cardano, le opere, le fonti, la vita, Milan: Francoangeli, 1999, pp. 169–179.

35 Although Bacon did not base his analysis of the separation of two plane surfaces on rarefaction, he did believe in this property of matter. See, for example, the description of air's rarefaction in the production of sound, Opus Majus, Part V, Chapter II.

Figure 0

Figure 1.Tartaglia's device. From Niccolò Tartaglia, Travagliata inventione, Venice: Nicolo Bescarini, 1551. Source: Mathematica Italiana website, URL: mathematica.sns.it.