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Galileo's new mathematical philosophy.

Abstract

Galileo's earliest mathematical work, on problems originating in literature and the arts, is closely related, thematically, to his late mathematical philosophy and the beginnings of modern physics. It is suggested that a key role in the transition from the early work to the late work, in which once seemingly isolated observations about the arts came to be seen as illustrations of a general and universal theory about the world, was played by one of Galileo's proteges, Niccolo Aggiunti, a person almost lost to history.

Keywords

Aggiunti, Galileo, philosophy of mathematics, scientific revolution

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The year 1610 marked a turning point in Galileo's career. Before 1610 he had been a fairly obscure, essentially unpublished, professor of mathematics at Padua. But in 1610, on the basis of the spectacular telescopic discoveries that he announced to the world in Sidereus Nuncius (Galileo, 1890-1909: III, parte prima), in which he dedicated the moons of Jupiter to the Medici as 'the Medicean Stars,' he became suddenly the most famous astronomer in Europe. Within a few weeks of publication, he was Court Mathematician and Court Philosopher to the Medicean Grand Duke.

His insistence in the correspondence leading up to this appointment that he be named not just mathematician but also philosopher had consequences that reverberated throughout the rest of his long life. No one could have objected to his being named Court Mathematician. Indeed, the term 'mathematician' was essentially equivalent to 'astrologer' in many contexts, a word that had even then a faintly comic and unwholesome connotation, but 'philosopher' was a title that carried the highest prestige. Unfortunately for Galileo he had no credentials as a philosopher whatsoever. On arrival in Florence he was immediately attacked on several fronts by real philosophers, that is, Aristotelian philosophers with university degrees in philosophy. The Copernican controversy is only the most notorious example of such an attack. The image of Galileo as an outspoken and combative figure derives from the period after 1610, when indeed his life was frequently a cycle of defense and counterattack, but all of this was arguably a distraction from what he was intending to do, and had been quietly doing even before 1610.

Galileo's life after 1610 is very well documented, but before 1610 the record is much sketchier. Researchers have sought evidence for Galileo's Copernicanism in this period, but in truth there is almost none. Although Galileo had to teach astronomy (the chief responsibility of a professor of mathematics) he seems not to have been particularly interested in it. A truly awkward moment occurred when he missed one of the most significant astronomical events of the millennium, the 'new star' of 1604 (the last supernova to be seen in our galaxy), first observed in Padua by his former student Baldessar Capra. Galileo only learned of it from a Venetian friend many days later. In his defense he wrote that Capra, as the first observer:
   should be held in appropriately high esteem, although one wonders
   whether those who aspire to some noble degree of glory in the
   mathematical sciences should spend every night of their lives in
   vigilant observation over the roofs of their houses to see if a new
   star might appear, so that others, who might have some advantage,
   should not have the triumph of such a glorious discovery.
   (1890-1909: II, 520) (1)


(This was less than three years before his first telescopic discoveries, when he began to do exactly what he ridicules here.)

An unbiased survey of the documentation of Galileo's life before 1610 would have to give special emphasis to mathematics, but not the mathematics of astronomy. Galileo's earliest education was in the Latin classics, drawing and music. According to his student Vincenzo Viviani, writing much later, he had been told that mathematics was the 'foundation' for the arts (Galileo, 1890-1909: XIX, 604). It is true that the Renaissance arts did have associated mathematical theories that claimed to be essential to their workings: perspective theory in the visual arts and numerical theories of tuning in music, for example. Galileo's father Vincenzo Galilei was even a prominent music theorist, thoroughly engaged with the mathematics of the arts, although not eager for his son to study mathematics. From about the age of 19, newly fascinated by mathematics, Galileo studied Euclid and Archimedes essentially on his own (having abandoned the study of medicine, his father's preferred career for him), a study that had nothing to do with astronomy. That he succeeded in gaining a university position and hence began teaching astronomy was a rather improbable accident, owing a great deal to his understanding of Archimedes and to a powerful patron, Guidobaldo del Monte, who admired some theorems that Galileo had sent him.

The place of mathematics in Aristotelian philosophy was respectable, but not particularly essential, in its role of filling in the details of what philosophy already knew about the heavens. Galileo's view of mathematics was different from this, but still his mathematics could not have had any clear philosophical meaning for him in 1610 when he suddenly embodied, in his own new titles, both mathematics and philosophy. Nonetheless, by the end of his life he had brought mathematics and philosophy together in a new way, with mathematics playing a much more essential role than it had ever done before. How that happened is the subject of this article.

The mathematics of proportion

In retrospect one can see that the notion of proportion is a thread connecting Galileo's early mathematics to his mature mathematical philosophy. The centrality and importance of proportion in his thought is clear in his very last (unpublished) work (Galileo, 1890-1909: VIII, 347-362), dictated in his blindness to Evangelista Torricelli, a projected sequel to his last book, Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Galileo, 1890-1909: Vili). Discorsi is a dialogue over four days, and this last work, which was intended to be a fifth day, is precisely on the concept of proportion, and specifically on Definition 5 of Book 5 of Euclid's Elements, which Galileo attempts to explain more simply than in Euclid's formulation. Definition 5 says what it means for pairs of unlike things to be in the same proportion.

The motivation in the dialogue for taking up this point is Galileo's celebrated Law of Fall, perhaps his most important theoretical discovery, treated on the Third Day of Discorsi. It says that for an object that falls from rest, speeding up as it falls, the speeds at two different times are in the same proportion as the times of fall. Galileo seems to feel that the meaning of this statement, being so fundamental, should be understood with complete precision.

A modern reader will almost certainly be puzzled that anyone could have misgivings about the notion of proportion. Reasoning by proportion seems to be a very simple thing. In Jean Piaget's theories of intellectual development, children around the age of 12 begin to reason by proportion. In a classic demonstration, a researcher shows a child a picture of a mother and a daughter. In the picture the mother is 6 inches high and the daughter is 4 inches high. Now the researcher shows an enlargement. In the enlargement the mother is 9 inches high. How high is the daughter in the enlargement? Children who haven't learned to reason by proportion think the child will still be two inches shorter, that is, 7 inches high. But older children, reasoning proportionally, correctly say 6 inches high. Both the mother and the child in the pictures have been enlarged in the proportion 3:2. What is difficult about this?

Euclid's Book 5 Definition 5 does not assume that we can put numbers on the quantities to be compared. Even if we could, the numbers wouldn't necessarily be integers, but rather irrational numbers, not suitable in any obvious way for doing arithmetic. Definition 5 is surprisingly subtle. The simplest proof of that is to note that for centuries it was not translated intelligibly from Greek. Euclid was available in Latin from the late Middle Ages, but Definition 5 (which for a while was Definition 6--further evidence of confusion) was gibberish. A definitively correct version did not appear until that of Federigo Comandino in 1572, in Galileo's own lifetime. The correct application of this idea in the arena of earthly physics was something new, and not at all trivial. No one worries about this issue now, but that is because the modern notion of the real numbers, finally put on a rigorous footing in the late 19th century (essentially by understanding, at last, the wisdom of Euclid's Definition 5), builds this concept into the way we do arithmetic.

If we look now at Galileo's early mathematics, we see that it is nearly always about proportion. The documentary record before 1610 may be sketchy, but it is quite rich in such examples. He even published a few of them as an appendix to Discorsi (although these are not usually reprinted in modern editions). A peculiar feature of these examples is that they derive not from physical problems but from literature. The new mathematical philosophy, in which mathematics had something to say about the world around us, was still in the future, but literature, especially stories with some sort of mathematical content, repeatedly furnished Galileo with mathematical problems.

The music of Pythagoras

A story of Pythagoras, told by Nicomachus of Gerasa (c. AD 60-c. 120) and repeated by Iamblichus (c. AD 245-c. 325), recounts how Pythagoras discovered the mathematics of music. Hearing the harmonious sounds of hammers on anvils in a smithy, Pythagoras weighed the hammers and found that their weights were in simple integer proportions. Experimenting further, he found that strings weighted in the same proportions vibrated in consonant musical intervals, like the octave (2:1), the fifth (3:2), and the fourth (4:3). This association of integer proportions with musical intervals became a dogma of Western music, transmitted to the Renaissance through the writings of Boethius (c. AD 480-524).

As it turns out, the consistent imposition of such integer ratios for all the musical intervals of the Western scale poses insoluble problems for tuning. Thus the strictest version of Pythagorean music theory contains, at bottom, a subtle contradiction.

The Galilei were, in the 16th and 17th centuries, a family of musicians. Galileo himself was a skilled lutenist, and his father, brother and a nephew were professional lutenists. His father Vincenzo was also a music theorist, and wrote prolifically on the problem of tuning in his Dialogo della musica antica, et della moderna (Galilei, 1581) and the Discorso intorno all'opere dimesser Gioseffo Zarlino (Galilei, 1589). The first book appeared when his son Galileo was 17, and the second when he was 25. In the intervening years Galileo had gone to the University of Pisa to study medicine, discovered mathematics and dropped out. Thus the second book, unlike the first book, is from a period when Galileo was a young adult at home, intensely occupied with mathematics. It contains, merely by the way, and not as part of its larger polemical argument about tuning, a remarkable observation about the Pythagoras story. Vincenzo says that he has found by experiment that the Pythagoras story in its usual form is not true. If you hang weights on strings to tune the strings, then to get the octave you need weights in the ratio 4:1, not 2:1, to get the fifth you need weights in the ratio 9:4, not 3:2, etc. That is, you must square the traditional proportions to get the proportions for the weights (Galilei, 1589: 104). It seems inconceivable that the young Galileo could not have been part of this discovery, being right there at the time.

He himself never describes experiments with plucked strings, but much later, in Discorsi, he summarizes these proportionality relationships in more detail than Vincenzo had stated them, including also, for example, the effect of making the string thicker. Interestingly, he omits other incorrect things that Vincenzo had said, for example that a resonant cavity must be increased in volume by the cube of the traditional ratio, 8:1, in order to produce the octave (Galilei, 1589:105): certainly not true.

In retrospect one can see this discovery as an experiment guided by mathematics, so close to what we now call physics that the result (a theoretical result nowadays) is part of every introductory physics course. At the time, however, it was just an odd observation about the Pythagoras story, one that Vincenzo merely mentions before returning to his polemic on tuning.

The Inferno lectures

Around the same time as the experiments on music, young Galileo, the university dropout, was invited to give two lectures on Dante's Inferno to the prestigious Florentine Academy (Galileo, 1890-1909: IX). This can only have been the work of Galileo's new patron, Guidobaldo del Monte, as a kind of inaugural event preceding his even more surprising appointment to the mathematics chair at his old university, Pisa. The poet Durs Grunbein, in an essay called Galilei vermisst Dantes Holle (Grunbein, 1996), ventures the inadvertently hilarious idea that these lectures represent the ultimate, definitive attack of science on the humanities. Grunbein appears not to know that Galileo was repeating arguments that were entirely within a Florentine humanistic tradition that, by the time it reached Galileo, was already 100 years old.

The consistent aim of this tradition was the glorification of Florence, and in particular the reclaiming of Dante for Florence, despite his invective against those who had banished him. Beginning with Cristoforo Landino's sumptuous edition of the Commedia in the late 15th century, the Inferno was studied minutely for every scrap of evidence that would help to trace the subterranean journey of Virgil and Dante, and in fact Dante does provide tantalizing evidence about this. Beyond the wood of the suicides, for example, there is a river that descends from the island of Crete, which must therefore be some hundreds of miles directly overhead. And confronted with the giants and Lucifer, Dante himself invites us to use proportional reasoning to understand their sizes and the sizes of the ice regions. The Florentine commentators elaborated theories about all of this repeatedly throughout the 16th century. The issue acquired an even more patriotic urgency when, in 1544, Alessandro Vellutello of Lucca published an edition of the Commedia that ridiculed the Florentine models of the Inferno, and introduced his own quite different one.

This controversy was the subject of Galileo's lectures, more than 40 years after Vellutello's publication. The lectures ostensibly present the Florentine Inferno in the first one and Vellutello's Inferno in the second, with the object of comparing them dispassionately, but of course the Florentine model is found to be far superior.

The Florentine Inferno is very grand, a conical cavity reaching from the center of the earth to a roof or cap, 400 miles thick, at the top. In contrast, Vellutello's Inferno is relatively tiny, just a small region near the center of the earth. One of Galileo's rhetorical strategies in the lectures is to belittle Vellutello's scheme for its ridiculously small size. He finds himself having to answer a contrary argument, however the suggestion that the roof over the Florentine Inferno, just 400 miles thick, would not be strong enough to avoid collapse. Imagine, he says, in dismissing this objection, a dome that is 30 braccia wide, as the roof over the Inferno is 3000 miles wide--it could be 4 braccia thick and it would certainly be strong enough to hold itself up. In fact it could be 2, or even just 1 braccia thick, and not 4, and would still be strong enough.

The dimensions that Galileo suggests here are not unlike those of Brunelleschi's dome, so that his argument is one that his listeners could assent to immediately. Soon after the lectures, however, Galileo must have noticed a spectacular flaw in this reasoning. The Brunelleschi dome is a scale model for the Inferno's roof, but what happens if you try to scale it up? The weight of the dome and the strength of the dome scale differently: one as the cube of the scale factor, the other only as the square (Galileo's own idea, published much later). That is, if you scale up by 10:1, the weight grows at 1000:1, but the strength only at 100:1. Relatively speaking, the strength is now less in the proportion 1:10. And using Galileo's own numbers here, the Inferno roof is weaker not just as 1:10, but as 1:100,000! It could never stay up. The Florentine model was impossible or worse, laughable, as proved by his own arguments!

For a young intellectual beginning a Medici-sponsored career it must have been an unexpectedly terrifying position. Galileo never admitted this mistake, but 50 years later he presented the scaling theory on the first day of Discorsi, as if glad to be done with it; not, of course, alluding to the Inferno, or in fact to anything architectural, although that would have been the most obvious application, but rather to the problems of supporting small and large ships under construction, or the relative strength of small and large animals. That is, he presented the theory, but changed the objects to which it was applied, making the origins of the idea all but invisible. Despite the innocuous uses to which the scaling theory is put in Discorsi, one of the interlocutors (Sagredo) describes himself feeling almost as if he had been struck by lightning to hear of this, and all the characters, including Salviati (Galileo's usual spokesman), admit that they had once thought you could scale things up arbitrarily. Another piece of evidence that Galileo hid this mistake, and was acutely embarrassed by it, is his failure to help a certain Luigi Alamanni find a copy of the lectures just a few years afterward, although he knew very well that there was a copy in his own hand with the Florentine Academy (Galileo, 1890-1909: IX). He quietly used the scaling theory in a different context, but without calling attention to it, in the 1612 publication alle cose che stanno in su l'acqua (1890-1909: IV), so he must have clearly recognized and corrected the problem by then. His last student, Vincenzo Viviani, who lived with Galileo and heard his stories, and collected all the documentation that he could find after Galileo's death, never heard about the Inferno lectures.

Like the experiments in music, the Inferno lectures and their aftermath show us the mathematics of proportion clarifying a point of physics. Today the scaling theory can be found in Chapter 1 of every introductory physics textbook. At the time, though, it was a most unwelcome surprise, not to be pursued or revealed.

La bilancetta

The story of Archimedes in the bath is found in Vitruvius' Ten Books on Architecture. According to Vitruvius, Archimedes helped his friend king Hiero of Syracuse detect the fraud of a goldsmith who had taken gold of a certain weight then given back a finished crown of the same weight, apparently gold, but actually gold mixed with some silver, so that the goldsmith was able to keep some of the gold for himself. How would Archimedes know this? As he realized while watching the water slop out of his bathtub, gold of the original weight, being very dense, would displace less water than an adulterated crown of that same weight.

In a little essay written around 1586, when he was 22, and never published in his lifetime, Galileo finds this story not credible (1890-1909:I). For one thing, he thinks the method would not even have the necessary accuracy. But, more importantly, he finds it simply not worthy of the god-like Archimedes. Rather, he suggests, Archimedes could have combined two apparently unrelated ideas from his known treatises to give a much more elegant and plausible solution to the problem of the crown.

In his treatise On Floating Bodies Archimedes proves that solids submerged in water seem to lose the weight of the water that they displace (Archimedes' principle). And in his treatise On the Equilibrium of Planes he proves that weights suspended from an ideal lever balance when the lever arms are in the same proportion as the weights (switched, so that the larger weight is on the smaller arm and the smaller weight is on the larger arm). Nowadays we would say that the torques balance. Galileo suggests combining these ideas as follows. An unknown substance is suspended on a lever in equilibrium with a counterweight on the other arm. Then water is brought up to submerge the unknown: it loses weight, so that it is no longer in equilibrium with the counterweight. The counterweight must be moved a certain distance (shortening the arm) to restore the equal proportion of arms and weights. The proportion of the distance moved reveals the proportion of weight lost, and that, in turn, is simply related to the density of the unknown substance. It is a device to measure density.

With this bilancetta one could determine the density of gold and then the density of the suspect crown to see if they are the same. This, he suggests, is what Archimedes must have done, and subsequent versions of the story, told by people who did not understand Archimedes' clever use of proportional reasoning, knowing only that he did it somehow with water, would have turned it into the childish version that we find in Vitruvius.

Galileo actually made a bilancetta and used it to determine the densities of many substances, including gems. His essay contains practical ideas for making and using it, apparently based on experience. A table of the values he obtained still exists (Galileo, 1890-1909: I), and he probably kept it for use throughout his long life because he characteristically refers to the density of this or that substance with confidence.

The table of experimental values is interesting for a peculiar reason. Many sub stances are represented twice, being measurements on two different samples, and the values do not exactly agree. This is probably an example of the inevitable uncertainty in measurement, although it might also be the case that the samples were not exactly the same pure substance. In any case it represents a very early confrontation with the disquieting difficulty of making mathematics, or even just numbers, apply to the natural world. The difficulty in doing this, the inevitability of 'measurement error,' had long been an unanswerable argument against any kind of terrestrial mathematical physics. Surely things in this sublunar sphere were subject to chance, to degeneration, to imperfection, to decay, to all the things that mathematics is not. Somehow Galileo was able to construct a useful table embodying mathematical law in which measurement error was explicitly present, without losing faith in the measurement. This subtle understanding is a necessary prerequisite for any quantitative experimental science, and Galileo managed it. One can perhaps point to his experience with the mathematics of the arts, where the criterion is not perfection but only the tolerance of the senses. A lute does not have to be tuned perfectly, for example, only well enough to satisfy the ear.

Alle cose che stanno in su l'acqua

An incident later in Galileo's life seems to have provoked him to publish a discovery connected with the bilancetta, and therefore presumably from this early period around 1586, although it only appeared in 1612 as the Discorsi alle cose che stanno in su l'acqua (Galileo, 1890-1909: IV), and then only under a kind of compulsion. He combines the same two Archimedean treatises that we have just mentioned, but in a completely novel way. From the ideas of On the Equilibrium of Planes Galileo gives a new derivation of Archimedes' principle, regarding floating as a kind of balance. In making and using a bilancetta, and thinking about the law of the lever as he watched the motion of the water, he would have noticed the phenomenon that underlies this new way of understanding Archimedes' principle. It is a remarkable argument, and must have confirmed for him the fundamental importance of the law of the lever, and more generally of the equality of proportions in nature.

The necessity of publishing this new derivation stemmed from one of the attacks that greeted him on arrival in Florence in 1610. Convinced of the truth of Archimedes' principle, he had remarked, in a philosophical setting, that ice must be less dense than water because it floats. In Aristotelian philosophy, however, ice was understood to be more dense than water, and it floated in spite of this because it was broad and flat. Several Aristotelian philosophers, notably the Florentine Ludovico delle Colombe, but also several professors at Pisa, attacked Galileo on this point, and delle Colombe even produced a chip of ebony that was more dense than water but nonetheless floated if carefully placed on the surface (because it was broad and flat, he said). A new theoretical derivation of Archimedes' principle is hardly a logical response to this challenge by experiment, but Galileo had it ready to use, and he produced it anyway. As we noted above, he also invoked the scaling theory at this point.

In revealing heretofore secret knowledge in a moment of crisis, Galileo was using a strategy that is not unfamiliar in Renaissance intellectual disputes. The most famous example is probably the solution to cubic equations, discovered in the late 15th century but kept secret for decades. Galileo was clearly keeping the scaling theory a secret for the eventuality of a situation like this one, and his new view of Archimedes' principle was completely beyond the abilities of his opponents to answer.

Motion

The examples above come, in essence, from very early in Galileo's career, or even before his career if we mean before his first professorship. All the ingredients for his mature mathematical philosophy can already be seen, embryonically at least, especially the use of proportion and his remarkable ability to interpret observation and experiment mathematically.

In his three years at Pisa he turned his attention to the problem of describing motion theoretically, especially accelerated motion, and he continued working on this problem in his 18 years at Padua, both theoretically and experimentally, laying the groundwork for what would eventually be the Law of Fall.

From 1609 on he put much effort into telescopic observations, showing again remarkable experimental skill in making his observations quantitative, and using his facility with proportional reasoning to interpret what he saw.

After 1610 Galileo became embroiled in many controversies, typically instigated by the enmity of Aristotelian philosophers. Already in 1610 Lodovico delle Colombe, one of Galileo's most persistent rivals, circulated a manuscript Contro il moto della terra aimed at Galileo (Galileo, 1890-1909: III, parte prima). We have Galileo's marginal comments in his own copy of this manuscript, and it is clear from these that Galileo's study of motion had led him to the profound discovery that we now call 'the Galilean principle of relativity.' As Galileo put it, 'motion in common is as if it did not exist.' (2) Arguments going back to Aristotle that seemed to prove that the earth must be at rest, essentially because if it were moving we would feel it moving and see the effects, were seen in light of the principle of relativity to be inconclusive.

Somewhat in tension with the principle of relativity was another conviction of Galileo's, traceable to his early Paduan days, namely that the tides are driven by the motion of the earth. (3) Neither the principle of relativity nor the theory of the tides is particularly mathematical. Galileo never tried to make his theory of the tides quantitative, and, as it turns out, it is not even qualitatively right (it seems to say that there should be only one high tide per day, not two). These ideas, whether right or wrong, are not mathematical, but rather physical, and open to interpretation. They made him uniquely vulnerable in the coming Copernican controversy.

As Galileo saw it, traditional arguments for a stationary earth were flawed, leaving it conceivable that the earth was moving after all. Furthermore, in his theory of the tides he saw the possibility of a physical proof that the earth really was moving. His opponents saw it differently. Most of them would have said that the tides were driven by the moon, so that one part of this argument fell flat. As for motion in common effectively not existing, the more sophisticated of his opponents agreed that we cannot tell physically if we are moving, so that we must have recourse to evidence of some other kind, like Holy Scripture, which does seem to say that the earth is at rest.

In his 1615 Letter to the Grand Duchess Christina (1890-1909: V) circulated in manuscript but not published, Galileo urged the Church to consider the possibility that the earth might be moving, and above all not to create a potential future embarrassment by linking Scripture to an assertion that might one day be proved physic ally false. His argument includes what was sometimes called the Book of Nature, namely God's own creation considered as a text that we might learn to read and understand, a work parallel, in a sense, to the Book of Scripture, but perhaps even to be preferred to Scripture in matters having nothing to do with our salvation, since the accommodation to the common understanding so necessary in Scripture would be quite unnecessary in the Book of Nature. Clearly Galileo had his own theory of the tides in mind as a possible proof that the earth really does move: more than just possible, that future embarrassment might be imminent! Nonetheless, in 1616 the Church declared the Copernican doctrine false, not least because it was 'foolish and absurd in philosophy.'

The years following this decree saw Galileo angry, withdrawn and ill. In 1618 three bright comets appeared that were observed carefully by the Jesuit astronomers of the Collegio Romano, but not by Galileo. He claimed to have been too sick even to go out to see them. Nonetheless, Galileo's friends urged him to write about them, and when he did it was an attack on the Jesuits, to their amazement. They had been his supporters in the early days, confirming his telescopic discoveries and honoring him highly on his 1611 visit to Rome. Because their founding documents committed them to Aristotle's philosophy, they had not been of any help in the events leading up to the condemnation of Copernicanism, but even then they seem to have been sym pathetic to Galileo's position. Now in an exchange of vitriolic publications Galileo alienated them completely, for no apparent reason. (4) His ideas in this exchange about the motions of the comets, the ones that he had not seen, seem strangely, even wilfully, confused.

Enter Niccolo Aggiunti

Niccolo Aggiunti (1600-1635) was a devoted disciple of Galileo. (5) Despite his very close, even affectionate, relationship with Galileo, he has never been accorded any particular importance by Galileo scholarship. The possibility suggested here, that Aggiunti is the key to Galileo's mature mathematical philosophy and a truly import ant figure in Galileo's life and work, must therefore be considered speculative. We will present evidence for this idea, portraying Galileo in a new and unexpected light: not alone but with a collaborator.

In order to separate speculation from what is actually known, we will depart slightly from a chronological account, circling back at the end to reconsider the critical period around 1621. What is certain is that in 1621 Aggiunti earned two laurea degrees at Pisa, one in philosophy and one in civil and canon law, and joined the Medici court, where he met Galileo. In 1626 Galileo made him professor of mathematics at Pisa (the chair he had once held himself). In 1633 Aggiunti is mentioned in a letter by Galileo's daughter Sister Maria Celeste: she had given Aggiunti and one of Galileo's in-laws the key to Galileo's house during his trial in Rome so that the two of them could remove possibly incriminating papers. On Galileo's return there was confusion about where those papers had gone, a source of acute, if temporary, embarrassment to Aggiunti. Aggiunti died young, and a memorial oration by the rector of the university, Marcantonio Pieralli (1638), is, with allowances for the floweriness of such compositions, another source of information about him. Antonio Favaro has written a short biography of Aggiunti (1913). Previously unidentified Aggiunti manuscripts are still occasionally discovered, and Michele Camerota in particular has written about these (Camerota, 1998). Most important, there are two volumes of Aggiunti manuscripts in the Collezione Galileana of the Biblioteca Nazionale Centrale di Firenze that have not been studied with anything like the attention that they deserve (Aggiunti, 1621?-1635, 1626?-1635).

Already in this bare account there is an implicit puzzle. Aggiunti was an Aristotelian philosopher from the very conservative philosophy department at Pisa, and Galileo's relationship with the Aristotelian philosophers could not have been worse when they met, yet they immediately became the closest of friends. None of Galileo's letters to Aggiunti survives, but there are many from Aggiunti to Galileo, beginning in 1624 when Galileo went to Rome to confer with Pope Urban VIII. We learn that in this short time Aggiunti had become, in effect, a member of the family. Aggiunti's tone in his first letter (Galileo, 1890-1909: XIII) could not be more familiar and affectionate if he were Galileo's own son, and the occasion of his writing was a reading aloud of two of Galileo's letters at the home of a Galileo in-law. This tone continues throughout Aggiunti's life. What is the basis for such intimacy?

Another puzzle is presented by Aggiunti's inaugural oration on assuming the mathematics professorship in 1626, published as a little book in 1627 with the title Oratione de Mathematicae Laudibus (Aggiunti, 1627). It asserts that mathematics underlies all the arts, all the sciences, all knowledge, and gives almost no special importance to astronomy, the usual sole application of university mathematics. Even more, it is an attack on conventional philosophy, portrayed here as a noisy, insubstantial disputing about nothing: when students of philosophy grow up,

they turn into tireless and chattering idlers. Thinking that knowledge is words ... they prefer rivers of words to a drop of thought, and markets, crossroads, and street corners daily ring with their earth shattering noise as they tie each other into knots with obscurities of riddles and enigmas, arguing with the shiftiness of Proteus or, like bankrupts who borrow to pay their debts, defending quibbles with quibbles. As they dispute with their captious arguments, their debate produces nothing of truth. Rather as Demonax the Cynic once said, 'One milks the billy-goat, the other holds the sieve.' ... By contrast, mathematical philosophers do not allow themselves to be carried away by rage ... but with minds calm and peaceful they ponder the problem; in difficult and deeply obscure matters they prefer rather to make an honest confession of ignorance than a rash assertion. (Aggiunti, 1627: 28) (6)

Who are these mathematical philosophers?

Although Galileo had not written to Kepler for 17 years, he immediately sent a copy of this little book to Kepler, on what must have been the very day that he received a copy from the printer. (7) In Kepler's view, despite his prodigious mathematical ability, the only worthy mathematical problems were astronomical ones. Galileo seems eager to recommend Aggiunti's oration to him, perhaps intending that Kepler should look at mathematics in this new way, namely that he should assign more importance to earthly problems. Mathematics, as exemplified in the arts, has been elevated in Aggiunti's oration to a level of philosophical importance that is entirely new.

It is clear that Aggiunti's appointment to the mathematics chair at Pisa was intended as the establishment of an alternative department of philosophy. In a 1627 letter to Galileo, (8) Aggiunti describes his students and his lectures, mentioning in particular his post-lezzioni. These were discussions of philosophical topics that would continue after his mathematical lectures. Many students, he says, hearing these, have turned away from Aristotelianism. Pieralli also mentions these posleziones in glowing terms in Aggiunti's memorial oration. (9) Another purely philosoph ical aspect of Aggiunti's mathematics professorship is an unpublished oration from the beginning of his second year of teaching. Known as De libertate philosophandi, (10) it is a virulently anti-Aristotelian attack on conventional philosophy, holding up Galileo's discoveries as proof that there is more for philosophy to discover. Although Aggiunti delivers it in his capacity as a mathematics professor welcoming his students back after the summer holiday, he hardly mentions mathematics at all. It is worth noticing that the university rector Marcantonio Pieralli seems to have been solidly behind this philosophical movement: an old friend of Galileo, he wrote the dedication to de Mathematicae Laudibus and saw it through the press.

A new Galilean mathematical philosophy had been installed at Pisa, but where exactly had it come from, and what exactly was it? The manuscripts left by Aggiunti might have some bearing on these questions.

The Aggiunti manuscripts

Manuscript collections Gal 128 and 129 of the Collezione Galileana of the BNCF are ascribed to Niccolo Aggiunti. Gal 128 is characterized as scientific manuscripts and Gal 129 as philosophical manuscripts. The oration De libertate philosophandi is in Gal 129, for example. The bulk of Gal 129, over 300 manuscript pages in Latin, seems to be a single sustained philosophical work in a rather old-fashioned style (syllogisms), reminiscent of the theological tradition of the High Middle Ages, with ubiquitous references to Aristotle, and many to St. Thomas Aquinas and Duns Scotus. Galileo's editor Antonio Favaro suggested that this cannot be a work of Aggiunti, and as he points out, it is not in Aggiunti's hand.

We are on firmer ground in Gal 128, which contains Aggiunti's writings without any doubt. Already in the 18th century GB Nelli noticed that Aggiunti had begun investigating the phenomenon that we now call capillarity, or, as Aggiunti called it, 'the occult motion of water.' (11) In fact, Aggiunti correctly considered together various phenomena that are all manifestations of surface tension, such as the shape of drops and their coalescence, the tendency of fluid to rise in a capillary tube and what that might imply for small animals such as mosquitos, the motion of water in plants and the absorptive property of sponges. It is worth noticing that already in de Mathematicae Laudibus he had suggested that there should be a mathematical theory for the way that plants draw water from the ground and convey it to their leaves (Aggiunti, 1627). The manuscript pages and paragraphs devoted to this occult motion are indicated in the margin by a wavy horizontal line, like the ruffled surface of a pond. One little drawing shows two communicating vessels filled with water, with one of the vessels just a thin tube. The water is higher in the thin tube, contra dicting the principle of communicating vessels, which says that the water levels should be the same, independent of the size of the vessels, a graphic manifestation of capillarity. The drawing appears to be an allusion to Galileo's alle cose che stanno in su l'acqua (1890-1909: IV), where Galileo gives a geometrical proof of the principle of communicating vessels with essentially the same drawing but with the levels equal.

Another topic in Gal 128, indicated in the margin of its pages by a drawing of a little tuning peg with a string in it, has to do with the mechanics of stretched strings. There are 28 propositions that develop the elasticity theory of strings: it has not previously been recognized how coherent and unified this section is. Aggiunti defines the notions that we now call stress and dimensionless strain, and demonstrates that, for a given substance, equal stress produces equal strain. He explicitly assumes that the stress-strain relation is nonlinear. This complication seems to be, ironically, a consequence of his skill as an experimenter: the gut strings that he experimented with apparently exhibited strain toughening, a phenomenon that is not surprising in a biological material. He makes creative use of the Galilean concept of 'momentum' (not our momentum: often closer to our 'gravitational potential energy'), an idea that he learned from alle cose che stanno in su l'acqua, and hence directly connected to Galileo's early mathematics.

This whole section seems to be motivated by a question that Galileo sent to Aggiunti from his house arrest in Siena in September 1633, along with a draft of the scaling theory that he had begun writing for Discorsi. Aggiunti refers quite excitedly to the scaling theory and to Galileo's question, whatever it was, in a letter of September, and then again in a second letter sent just a week later along with some kind of response to the question (Galileo, 1890-1909: XV). We can deduce that the question was 'whether long strings are more apt to break than short ones,' because Aggiunti answers this question twice, once following his Proposition 9 and then again following his Proposition 28, the result of going back and reworking the material of Propositions 1-9 and adding a great deal more, including a theory of the vibrations of stretched strings. In Discorsi Galileo incorporated Aggiunti's idea about how strings break in a very quick digression from the scaling theory (1890-1909: VIII, 162).

Aggiunti's view of how strings stretch is motivated by a microscopic picture: he imagines the string to be a chain of small stretchable units that he calls 'syphunculi.' Each of them stretches by the same small amount, and what we see macroscopically is the sum, the total accumulation of all these small stretches. Galileo took only the idea that if one small part of the string should fail, it was due to the force on that part alone, the rest of the string being irrelevant. Aggiunti does something more interesting though. Immediately after motivating his theory with the microscopic syphunculi, he does away with them, geometrizing the theory using Euclid's Definition 5 of Book 5. Equal strain will now mean not that each syphunculus is equally stretched, but rather that each part of the string increases its length in the same proportion. This is an unmistakeably Galilean touch, especially in light of the projected Fifth Day of Discorsi.

It is clear, then, what the new mathematical philosophy was: the application of geometrical abstraction, geometrical reasoning and geometrical representation to observable phenomena and experiment. Galileo does it in Discorsi, but he was not alone. Aggiunti did it too. The two of them were in very close touch about this kind of philosophizing and seem to have understood each other perfectly.

Gal 129: The philosophical manuscripts

The puzzling aspects of this collaboration are somewhat resolved if we imagine that when Galileo and Aggiunti met in 1621 they discovered a mutual interest in the philosophy of mathematics. Aggiunti was a philosopher, after all, not a mathematician. Together they might have come to a formulation that gave mathematics its new philosophical importance, the kind of universality that is expressed a few years later in de Mathematicae Laudibus. The philosophical conviction that mathematics provides a way to reason about the world around us would have given Aggiunti the confidence to investigate stretched strings in the geometric way that he later did. Galileo could have made Aggiunti professor of mathematics in 1626 over another protege, Cavalieri, who also wanted the job, not because Aggiunti was a better mathematician (he wasn't), but because the new philosophy of mathematics was the whole point.

Although Antonio Favaro has suggested that the long philosophical treatise in Gal 129 is not by Aggiunti, there is another possibility: it could be Aggiunti's laurea thesis. That would explain the presence of this apparent anachronism among his papers in 1635. It occupies the bulk of Gal 129, c. 47-c. 220, over 300 pages of Latin manuscript. It is a nearly complete treatise in the philosophical tradition of Aristotle and St. Thomas Aquinas. The author of this treatise typically favors Aquinas over Duns Scotus when they conflict, but he is not at all afraid to stake out his own positions.

Favaro has pointed out that when Aggiunti's papers were collected from his house after his unexpected death, papers that just happened to be in his possession could have been mixed with his own papers and thus found their way into Gal 129. This is true, of course, but it is arguably even harder to understand what Aggiunti was doing with them in 1635 if they were not his. After all, he was a laureate from a notoriously conservative philosophy department, and canon law requires the knowledge of just such topics and methods of disputation as we find in the Gal 129 treatise. The audacity and comprehensiveness of the work, which even reconsiders such foundational subjects as grammar, is not at all inconsistent with Pieralli's description of Aggiunti as a student (Pieralli, 1638). Its sprawling text contains sections, in order, called 'An Universam Aristotelis Logicam Questiones Prohemiales' (c. 47r-c. 89v), 'Logica Brevis' (c. 90r-153r) and 'Disputatio in Uni versali ter' (c. 153v-c. 196v), although the Logica Brevis turns out to be part of the Prohemiales. The end of the Disputatio in Universaliter is missing, as is the beginning of the last section (c. 197r-c. 220v) which therefore is missing its title, although it appears to have been 'Tractatus de Anima.' (12) I will call this whole text simply Aggiunti's thesis.

In dismissing Gal 129 Favaro rightly points out that it is not in Aggiunti's hand, but in fact it is not in any one person's hand. Much of the thesis has been serially copied from many notebooks that were designated by number, typically one copyist for one or two notebooks, and the numbered source for each part of the thesis is indicated at the beginning of each such part. This is one way to see that the thesis is fairly complete. Books 2, 3, 5, 6, 7, 8 and 9 are represented in order, covering pages c. 101r-c.196v. Book 4 might seem to be missing, but in fact the number of pages from 3 to 5 is just right for two notebooks, because they were evidently octavo notebooks, the same size as the ones that Aggiunti used, and that still exist, in Gal 128. The copyist who copied books 3 and 4 neglected to indicate where book 4 began. Both Question 7 of Logica Brevis and the end of Logica Brevis make direct reference to the beginning of Prohemiales on c. 47r, so continuity from c. 47r-c. 196v is clear. There is a break at c. 196v, where material has certainly been lost, and the remaining text, the incomplete 'Tractatus de Anima,' is not so obviously part of what came before, although it was copied from notebooks 13 and 14, consistent with being the continuation, but with notebooks 10-12 lost. It ends with 'Final Question: Whether the rational soul is immortal,' (13) which certainly sounds like the end.

If Aggiunti authored this thesis, then we have a rather detailed picture of the Aggiunti that Galileo met in 1621. What would they have had to talk about? In 1621 Galileo was unquestionably angry about the prohibitions of 1616 and he was becoming ever more enmeshed in the controversy on comets. He was writing II Saggiatore, an angry book. One might not expect him to get on well with a brash young Aristotelian philospher.

But in fact what Aggiunti represented in Aristotelian philosophy had virtually nothing to do with physics or cosmology, the places where Galileo had run afoul of Aristotelianism. The Thomastic synthesis of Aristotle and Christian theology included much more than Physics and De Caelo. Aristotle had asked how we know anything, how we express what we know and a thousand other things. Two crucial concepts in the Christian/Aristotelian synthesis, central to Aggiunti's thesis, were 'singularities' and 'universals,' the first localized in space and time, the second eternal and everywhere. What do we know of singularities, what do we know of universals and how are they related? We might find these questions more familiar if we asked 'how do experiments (singularities) tell us about laws of nature (universals)?' These were questions that Aggiunti was well prepared to discuss.

Aggiunti's thesis seems to formulate a synthesis of Aristotelian logic with Aristotle's de Anima in the context of Christian scholastic tradition. In particular, his theory of logic, formulated in the Prohemiales, divides logic into two parts that have their counterparts in the theory of the soul: Natural Logic and Artificial Logic. Natural Logic is effortless and includes, for example, our sensory systems. Artificial Logic is conscious, rigorous, mental effort that works with the information provided by the senses. We humans need Artificial Logic to arrive at true knowledge, because human Natural Logic is too weak to arrive at true knowledge by itself. Angels, on the other hand, do not need Artificial Logic, because they instantly perceive the truth through Natural Logic. Angels aside, one can see in this doctrine things that might have interested Galileo. The philosophy of knowledge developed here and in this whole tradition was aimed at theological knowledge, but it was couched so generally that it could be interpreted just as well with respect to other kinds of knowledge, including scientific knowledge.

Galileo had argued in vain the validity and importance of reading the Book of Nature in the context of the Copernican controversy, but here was what looked like a Christian/Aristotelian description of that process, and even a detailed theological justification for it. Galileo could have simply understood Artificial Logic as another name for mathematics, and it would be just what he had been saying himself.

For Aggiunti, though. Artificial Logic was not mathematics. Mathematics (per haps associated in his mind with Pythagoras) was more like an earlier, primitive part of Artificial Logic. He credits Aristotle as the inventor of Artificial Logic, and points out that earlier philosophers had made errors in logic, such as Socrates, who con fessed 'the one thing that I know is that I know nothing.' (14) (I do not think that Aggiunti was joking here.) Galileo and Aggiunti would have disagreed on the nature of mathematics, but this was just the kind of dispute that both of them loved and were good at. Aggiunti would not have given up easily. The valorous defense of his own ideas against the erudite and subtle arguments of his opponents was just what he was famous for, according to Pieralli. But ultimately Galileo won him over. That is what De mathematicae laudibus is: the final assertion that mathematics is the Artificial Logic that we use to turn sense perception into vera scientia.

They also would have disagreed on the nature of the senses. This is the content of the incomplete last section of the thesis. From what is there we can see that Aggiunti subscribed to the Aristotelian idea that our sensory systems receive detailed information about what is beyond us, and that our conscious perceptions are precisely this information. Galileo argued that what is beyond us does not have all the qua lities that we perceive, but rather our sensory systems add many of these qualities, which are therefore in us and not outside us. Once again we can find the outcome of their dispute in De mathematicae laudibus, where Aggiunti says:
   You may range over the sphere of the stars and the earth, you may
   journey in your mind through the whole universe of creation--in all
   things whose appearances reach the mind through the senses, what do
   we have, what do we see except motions, colors, numbers and shapes?
   Nothing, of course. If we want to get to know anything, we have to
   concentrate entirely on these features and exercise our brains on
   them alone; but it is only geometry that considers them carefully
   and investigates them thoroughly. (15)


This is a complete concession to Galileo's point of view on the senses, but the role asserted here for geometry is new for both of them.

They met when Galileo was writing Il Saggiatore, and that book contains unexpected philosophical digressions from its overall polemical theme on just the sorts of topics that Galileo and Aggiunti would have been discussing. These digressions are so striking that one almost forgets that the book is supposedly about the comets of 1618. There is, for example, a passage of several pages on Galileo's theory of the senses (1890-1909: VI, 347-352), although it seems to have nothing to do with the comet controversy. There is the famous 'manifesto' on the universality of mathematics, something that he had not really said before:
   Philosophy is written in this grand book, the universe, which
   stands continually open to our gaze. But the book cannot be
   understood unless one first learns to comprehend the language and
   read the letters in which it is composed. It is written in the
   language of mathematics, and its characters are triangles, circles,
   and other geometric figures without which it is humanly impossible
   to understand a single word of it; without these, one wanders about
   in a dark labyrinth. (16)


And there is the odd parable of the man who sought to know how sound was produced. It is meant to be a lesson in humility to someone who should know that he does not know. But it is oddly like Aggiunti's conception of someone who has only Natural Logic and not Artificial Logic. The man in the parable takes in sensory information, and learns all kinds of ways that sound is produced by looking and listening, but he never thinks about it in any effective way, so he is continually surprised when anything changes. He is 'wandering about in a dark labyrinth,' to quote the manifesto above. Eventually he concedes that what he knows is effectively nothing. This is just Aggiunti's idea of the weakness of the human intellect in the absence of Artificial Logic. It is a disquieting story because we know that Galileo is not advocating mindless acceptance of whatever comes and T do not know' as a final answer. Urban VIII reportedly loved this story though. Perhaps the resonance of its theological origins came through.

In such passages Aggiunti may have contributed, indirectly at least, to some of Galileo's most memorable formulations.

In their disputes Aggiunti and Galileo became true friends, as Aggiunti's letters prove. They also must have reached an understanding of a new philosophy of math ematics, one that gave it the vast new importance of Aggiunti's Artificial Logic, quite unlike mathematics' traditional role. In this context there could be no question of appointing Cavalieri to the mathematics chair at Pisa. Cavalieri was just a good math ematician, but Aggiunti was uniquely able to convey a new conception of mathematics.

The death of Aggiunti was the beginning of the end of the new philosophy, except to the extent that it was carried on elsewhere and merged with developments in other places to become modern science. Aggiunti's best friend Dino Peri was appointed to replace him, but there is no sign that Aggiunti's abilities had rubbed off on Peri. A young mathematics prodigy, Vincenzo Viviani, was brought to live with Galileo, possibly to be the new Aggiunti, so to speak.

Indeed, Viviani was later to become a stalwart member of the Academia del Cimento, the scientific society founded by Ferdinand II and his brother Prince Leopold in an attempt to continue Galileo's work, although they did not completely understand what that work had been. It is interesting to note that among Viviani's writings is a treatise on Euclid's Book V, Definition 5, but the members of the Cimento itself explicitly gave up on geometrical theorizing and restricted themselves to observation and experiment, as they state in the introduction to their one and only publication, the Saggi. Without mathematics their years of experimental work had, alas, no repercussions at all. It was left to Isaac Newton in his Principia Mathematica to cite Galileo's Law of Fall as the first and clearest example of his own new mathematical philosophy.

DOI: 10.1177/0014585815570830

Acknowledgements

I wish to express my gratitude to Michele Camerota for guidance in reading Aggiunti, and to Marco Beretta, Thomas Settle, and the staff at Museo Galileo for kind hospitality.

Funding

This work was supported in part by a Faculty Grant from Mount Holyoke College, USA.

References

Aggiunti N (16217-1635) Philosophical manuscripts (unpublished), Gal 129. BNCF.

Aggiunti N (16267-1635) Scientific manuscripts (unpublished), Gal 128. BNCF.

Aggiunti N (1627) Oratione de Mathematicae Laudibus. Rome: Mascardi.

Camerota M (1998) Adattar la volgar lingua ai filosofici discorsi. Una inedita orazione di Niccol Aggiunti contro Aristotele e per l'uso della lingua italiana nelle dissertazioni scientifiche. Nuncius 13(2): 595-624.

Favaro A (1913) Amici e corrispondenti di Galileo. XXX. Niccolo Aggiunti. Atti del R. Istituto Veneto di scienze, lettere ed arti, Voi. LXXIII, Part 2, 1913-1914, pp. 1-77 (reprinted in Favaro A (1983) Amici e corrispondenti di Galileo, a cura e con nota introduttiva di P Galluzzi, pp. 1167-1243). Florence: Libreria Editrice Salimbeni.

Galilei V (1581) Dialogo della musica antica, et della moderna. Florence: Giorgio Marescotti.

Galilei V (1589) Discorso intorno all'opere di messer Gioseffo Zarlino. Florence: Giorgio Marescotti.

Galileo (1890-1909) Le Opere di Galileo I-XX, edizione nazionale sotto gli auspicii di Sua Maest il Re d'Italia. Florence: Tip. di G Barbra.

Grunbein D (1996) Galilei vennisst Dantes Holle. Berlin: Suhrkamp.

Pieralli MA (1638) Orazione [...] recitata pubblicamente [...] in memoria dell'Eccellentiss. Sig. Niccolo Aggiunti. Pisa: Francesco della Dote.

Mark A Peterson

Mount Holyoke College, USA

Notes

(1.) '1 qual primato se deve esser tenuto in cosi grande stima, sara bene che quelli che nelle scienze matematiche aspirano a qualche nobil grado di gloria, trapassino tutte le notti della lor vita in osservar con gran vigilanza sopra i colmi delle case loro se qualche nuova stella apparisce, accio che altri, a i quali il caso fusse piu favorevole, non riportassero la palma di cosi glorioso scoprimento.'

(2.) The Second Day of Galileo's ill-fated Dialogue Concerning the Two Principal World Systems is devoted to the principle of relativity.

(3.) The Fourth Day of the ill-fated Dialogue is devoted to the tides.

(4.) This exchange occupies most of Volume VI (Galileo, 1890-1909).

(5.) Antonio Favaro (1913) calls Aggiunti 'un nato discepolo.'

(6.) Aggiunti (1627: 28): 'scientiam verba putantes ut lucum Ugna, praeoptant verborum flumina quam mentis guttam, atque his mundi cymbalis quotidie fora, compita, trivia perstrepunt, dum tricarum griphorumque perplexitatibus se mutuo implicantes, & versi pelli tergiversatione altercantes, aut decoctorum more, qui versuras versuris solvunt, distinctiones distinctionibus tuentes captiosisque argutationibus contendentes nihil dis putando veri exprimunt, sed ut quondam Demonax Cynicus, alter hircum mulget, alter supponit cribrum ... Contra vero philosophi mathematici non se ira, quae artem pertur bat, ut Pyrrus aiebat transversos abripi finunt, sed totos se rationi permittunt, non obstin ate refractarii iurgiis & conviciis invehuntur, sed pacatis sedatisque mentibus rem perpendunt, & in obscuris poenitusque abditis afferunt potius ingenuam ignorationis confessionem, quam temerariam affirmationem.' (Translation by Philippa Goold.)

(7.) Postscript, letter to Kepler of 28 August 1627 (Galileo, 1890-1909: XIII).

(8.) Letter of 16 May 1627 (Galileo, 1890-1909: XIII).

(9.) Quoted in Favaro (1913).

(10.) The first of the Aggiunti manuscripts in Gal 129 (16217-1635).

(11.) Cited in Favaro (1913: p. 48, n. 4).

(12.) Gal 129, 207r.: 'Scopus totius tractatus de Anima est precipue ut dare concludatur ...'

(13.) Gal 129, 207r.: 'Questio Ultima: an anima rationalis sit immortalis.'

(14.) Gal 129, 52r.: 'Socrates fassi sunt se hoc unum scire nihil scire.'

(15.) Aggiunti (1627: 8): 'Astrorum, terrarumque orbem cogitatione pervagemini, versatilem hanc rerum universitatem animo perlustretis, ex his rebus omnibus, quarum species sensuum ministerio ad animum pervadunt, quid nos habemus, quid percipimus praeter motus, colores, numeros, ac figuras? nihil profecto. Si ergo ullius rei notitiam consequi velimus, ad haec omnino mentem convertere opus fuerit, et super his dum taxat ingenium agitare; sed sola Geometria haec curiose contemplatur, et sagacissime pervestigat.'

(16.) Galileo (1890-1909: VI, 232): 'La filosofia e scritta in questo grandissimo libro che con tinuamente ci sta aperto innanzi a gli occhi (io dico l'universo), ma non si puo intendere se prima non s'impara a intender la lingua, e conoscer i caratteri, ne'quali e scritto. Egli e scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi e impossibile a intenderne umanamente parola; senza questi e un aggirarsi vanamente per un oscuro laberinto'.

Corresponding author:

Mark A Peterson, Mount Holyoke College, 50 College Street, 423 Clapp Laboratory, South Hadley, MA 01075, USA.

Email: mpeterso@mtholyoke.edu
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