PETER MACHAMER

2         Galileo's machines, his mathematics, and his experiments1

Galileo's life and works, like Gaul, has generally been divided into three parts. There is his work on mechanics and local motion (with the science of the strength of bodies grudgingly admitted), his work on astronomy and Copernicanism, and, finally, his relations and tribulations with the Catholic Church. Occasionally, some scholars under the anachronistic rubric of methodology have attempted to tie the two scientific parts together to obtain a more coherent picture.

In the space of this essay, I cannot overcome this tripartite division that the centuries have sanctioned. I can, however, sketch a picture of Galileo that will be a step toward this goal. It seems to me that Galileo had only a few basic conceptions that directed his life and work in all these three areas. His first belief concerned the role of the properly thinking and working individual scientist as being able to obtain knowledge and certainty. This belief showed up in his writings about the role of the scientific elite individual, who saw and understood things that were not seen by the masses, or, especially, by groups dedicated to the authority of Aristotle. It also seems to lie at the heart of his thoughts about Biblical exegesis. This individualism is apparent in Galileo's texts by his ubiquitous and consistent use of the first person singular “I” (“io”) and the use of proper names or descriptions to talk about insights, discoveries, and all the accomplishments of good science. I call this the entrepreneurial-I. In Il Saggiatore, Galileo wrote eloquently on this individualistic theme:

Sarsi perhaps believes that all hosts of good philosophers may be enclosed within walls of some sort. I believe, Sarsi, that they fly, and that they fly alone like eagles, and not like starlings [storni]. It is true that because eagles are scarce they are a little seen and less heard, whereas birds that fly in flocks fill the sky with shrieks and cries wherever they settle, and befoul the earth beneath them. But if true philosophers are like eagles, and not like the phoenix instead, Sig. Sarsi, the crowd of fools who know nothing is infinite; many are those who know very little of philosophy, few, indeed, they who truly know some part of it, and only one knows all, for that is God.2

Tempering this vision of the entrepreneurial scientist, a moderate egalitarian character also emerges. Galileo sometimes seems to believe that anyone receiving proper tutelage and paying proper attention can learn what is true. This is his homage to the Platonic doctrine of recollection (anemnesis). The emphasis on individualism also is used to make the typical Renaissance points. Galileo was much in the “modern” tradition of being extremely anti-Scholastic, anti-Aristotelian, dogmatically antiauthoritarian, and, somewhat uniquely for an Italian philosopher, antioccult. However, I will not go further into how Galileo fits in with the seventeenth-century rise of epistemic, economic, and political individualism.³

I will concentrate on some aspects of what Galileo took to be intelligible and the model of intelligibility that he developed (or constructed, if you prefer.) This second theme is related to the structures of nature and how truth comes to be known and displayed in natural philosophy. I will argue that Galileo's model of what was intelligible comes from Archimedes and the simple machines. This model was, and was seen to be, a new philosophy of nature, and it provided a new model for subsequent generations of how to do natural philosophy. This is not to say that Galileo had no predecessors. Of course there were many.⁴ But no one, until Galileo, made the mechanical way, the way of doing science, the way of knowledge.

The story has yet to be told of how Galileo, who was probably not the brightest nor the best of the mechanicians nor of the mathematicians nor of the philosophers, was the one who made the mathematical, mechanical way the future of science. One can speculate that his successes and international acclaim with the telescope made him Europe's popular and intellectual hero, and so people were more likely to take seriously what he said on other topics. Maybe it was because he was extremely smart about the topic of motion, and his work predated his telescopic period, and because studying the motion of things was in accord with the tenor of the times in claiming novelty and some historical precedent. Maybe it was because this mechanical way presented, even more than Galileo saw, an alternative, systematic approach to dealing with the world.5

His way of stating and solving problems in natural philosophy in mechanical ways became the model of natural philosophy for the seventeenth century. After Newton's work (solidified by Euler) things would change. No longer would statics, rational mechanics, and dynamics be taken to be the same discipline. Further, proportional geometry would be supplanted by algebra. This made the “new” science even newer.

From the 1930s through the 1960s much of the debate about the nature of early modern science revolved around attributing the labels of Platonism and Aristotelianism to various practitioners.⁶ By and large these Plato-Aristotle debates were centered on the concept of the proper scientific method. The general lines were that Aristotelians went back to the Posterior Analytics and experience and, therefore, experiment, whereas the Platonists made use of mathematics. So Alexandre Koyré characterized the Renaissance debate between Aristotle and Plato by claiming that if a thinker believed in the descriptive power of mathematics, he was a Platonist.⁷

But obviously the lines dividing different practitioners were not so clear. Though we know that Albertus Magnos unmasked the liber de causis as not being the work of Aristotle, about two and a half centuries later, Francesco Patrizzi issued a Latin edition of the Elements of Theology under the name of Aristotle. Similarly, even before Koyré had characterized Platonism as mathematics, Ernst Cassirer had found over sixteen distinguishable types of Platonism in the sixteenth century.⁸

There were debates in the late sixteenth century over whether certitude was better attributed to mathematical or syllogistic reasoning, but these were not held under the guise of Plato and Aristotle, and Petrus Catena, in the mid sixteenth century, was busy in Venice publishing works that showed Aristotle's important use of mathematics.

Mathematics itself came in many guises both institutionally and extrainstitutionally. Certainly, geometry was taught at the universities, but also there were the mathematical sciences of astronomy, geography, and sometimes mechanics. Outside the sanctioned institutions mathematics reigned quite lively in the realms of natural magic, astrology, hermetic practices, and the cabala, as well as in the more mundane, pragmatic spheres such as the principles of painting, construction of fortifications, and the design of machines.

What is to be learned from the historical complexity concerning the earlier historiographic Plato-Aristotle debate? Basically this: The categories of the historian's debate were not actor's categories. They were anachronistically imposed structural categories used by historians in an attempt to bring some order to the complexity. Most often, the order imposed was related to the historian's vision of the correct nature of modern science.

It is arguable, and reasonably so, that virtually any historical study is anachronistic in some degree. Every historian must be selective, choosing his characters, texts, and institutions with an eye to what is deemed important. In chronological histories, the issue of precursorship looms large, and so everyone exhibits some degree of Whiggish prejudice.

Yet, as has been argued from Burkhardt onward, categorization is both useful and necessary in history. Similarly it is useful and necessary in psychology to ascribe categories for understanding how peoples, at any time, think and make sense of their world, what they use to make intelligible sense out of their experiences.

But the categories depicted by the names “Plato” and “Aristotle” by themselves as historical personages do not seem today to be of much help in understanding the precursorship of modern science. Certainly at the end of the sixteenth century there was a long-lasting and multicomplex Aristotelian tradition and a renewed and resurgent, somewhat oppositional, Platonic tradition (for example, Bologna had established a chair in Plato by the mid-sixteenth century), and certainly many forms of systematization were floating about in various degrees of completion that could be called neo-Platonic. But even these were mostly Christian hierarchical bastards designed for a variety of religious, educational, and political purposes.

Despite these categorical caveats, I think at least one other name needs to be added to the list of categorical types for understanding the development of modern science: Archimedes. Recently Olaf Pedersen made a similar suggestion, when he suggested seeing the development of scientific method in terms of three great traditions: Plato, Aristotle, and Pythagoras or Archimedes.9 But I want to go somewhat further. Of these three, I would claim that in the early seventeenth century, only the name of Archimedes invoked the original vision of the person denominated.

Surely, those who followed Archimedes were also often Christians and held a wide variety of beliefs in addition to their Archimedian vision. But I will claim that the scientific revolution of the seventeenth century owes much to an almost pure form of the Archimedian mathematics. It is this pure form that gave rise to the mathematical and experimental structure of the mechanical world picture (though not to all aspects of its widespread power of intelligibility). There were other helpful models that captured the imagination of the seventeenth century and aided the acceptance and spread of this mechanical view, most notably the image of the mechanical clock. But the “guts” of the picture comes from Archimedes, and almost solely originally through one long sung “hero of science”: Galileo. So today let me sketch for you what I take to be a new view of Galileo and his Archimedianism.10

MACHINES: GALILEO AND THE BALANCE

Much has been written about Galileo's artisan-engineering training. Leonard Olschki, Erwin Panofsky, Lynn White, and Tom Settle all have drawn attention to this aspect of Galileo's background.¹¹ Wallace Hooper most insightfully developed another aspect.¹² I think that this artisan story is right, but I want to treat it differently and make more of it than they do. I argued many years ago that Galileo belongs in the tradition of the mixed sciences (scientia media).¹³ In effect, this means that his use of the traditional Aristotelian causes is specialized to cases where the phenomena can be seen as both physical and mathematical. So, for example, formal causes and efficient cause often collapse. In this essay I want to draw further implications from one type of mixed science, mechanics. I will argue that the Archimedian simple machines and the experiences related to them become Galileo's model both for theory and for experiment. William Wallace agrees with his mixed science perspective, and yet, he emphasizes Galileo's debt to the philosophy professors in the Collegio Romano.¹⁴ But then the question is how do these two types of training and influence relate to one another.

The short summary is that Galileo is an Archimedian mechanic by training and temperament, working in the mixed science tradition, and desperately trying to advance himself intellectually, socially, and financially by seeking legitimacy as a philosopher. For this, he must use acceptable scholastic terminology and deal with the problems of traditional natural philosophy. So he tries to apply his mechanical interests and insights to Aristotelian and peripatetic questions of natural philosophy, and he makes every effort to use their accepted mode of philosophical speech.

This attempt first becomes clear in his De Motu (Galileo 1590). Galileo wrote De Motu (On Motion) in the early part of his career; the traditional date is 1590 (by Favaro and Drabkin, but see Hooper.15) Whatever the exact date of composition, the manuscript was written while Galileo was teaching at Pisa. It was never published. However, it stands as an early example of Galileo's model of good science, despite the basic mistake of treating the motive power of bodies as being the relative difference between their specific gravity and that of the media in which they are immersed. His discomfort, perhaps based on this mistake, may be the reason De Motu was never published. By 1604 in his letter to Paolo Sarpi he seems to have abandoned this mistaken conception.¹⁶

In De Motu Galileo attempted to show the inadequacies of the Aristotelian theory of natural motions (where it was held that the heavy and the light were two different causes of motion). He argued in favor of a unified causal theory of natural motion, where what needs to be known is the proportional relation of the weight per volume of a body (conceived as the force caused by the weight) to the weight per volume of its surrounding medium. This is Galileo's way of describing the forces that act upon the body to make it go up, down, or remain at rest. Put another way, he first transformed problems about falling bodies (freef all) into a problem of hydrostatics (floating bodies) in which the body is seen to be rising, falling, or floating in a medium.

He then argued that all these phenomena (falling, floating, etc.) should be seen as balance problems, so he titles his ninth section: “In which all that was demonstrated above is considered in physical terms, and bodies moving naturally are reduced to weights of a balance.”¹⁷ This balance model for solving motion problems he credited to Archimedes.

Galileo argued at greater length in Chapter 6 that “what moves, as it were by force”¹⁸ and showed how “the motion of bodies moving naturally can be reduced to the motion of weights on a balance.”¹⁹ The section title reads: “In which is explained the analogy [convenientia] between naturally moving things and the weights on a balance.”20 Interestingly, he says this is a physical, as opposed to a mathematical, argument:

We shall first examine what happens in the case of the balance, so that we may then show [ostendamus] that all these things happen in the case of bodies moving naturally.

Let line ab, then represent a balance, whose center, over which motion may take place, is the point c bisecting line ab. And let two weights, e and o, be suspended from points a and b.

(Fig. 2, Balance, De Motu, [NE 1 257].)

Now in the case of weight e there are three possibilities: It may either be at rest, or move upward, or move downward. If therefore weight e is heavier than weight o, then e will move downward. But if e is less heavy, it will, of course, move upward, and not because it does not have weight, but because the weight of o is greater. From this it is clear that, in the case of the balance, motion upward as well as motion downward takes place because of weight but in a different way: For motion upward will occur for e on account of the weight of o, but motion downward will occur for e on account of its own weight. But if the weight of e is equal to that of o, then e will move neither upward nor downward.²¹

Next Galileo returns to the case of naturally moving bodies and tells us that a volume of water equal to a volume of wood will be heavier, which is why the wood cannot be submerged beneath the water. In general, he claims, all things can be explained in the same way, which of course disagrees with Aristotle:

In the case of bodies moving naturally, as in the case of the balance, the cause of all motions up or down can be referred to weight alone … what is moved is moved, as it were by force, and by the extruding action of the medium.²²

This anti-Aristotelian conclusion about the nature of motion, all being caused by weight, is most important because all natural motion (sublunary) is reduced to this one cause.

He repeats this same claim with further explication of the identity between floating bodies and the balance in Chapter 9:

Let us consider how and why bodies moving upward move with a force measured by the amount by which the weight of a volume of the medium (through which motion takes place) equal to the volume of the moving body exceeds the weight of the body itself … [And later] For if the weights [on a scale] are in balance, and an additional weight is added to one side, then that side moves down, not in consequence of its whole weight, but only by reason of the weight by which it exceeds the weight on the other side.²³

The principle to be noted is that for Galileo the whole schema of intelligibility becomes putting a question in the form of an equilibrium problem: What is the cause of (or force that causes) something becoming unbalanced? And what force will cause it to come back into balance? Where is the balance point? The geometrical diagrams Galileo used to represent these problems were always lines and angles inscribed in, circumscribed about, or tangent to circles. These literally described a real balance but also allowed him to use rules of geometry for constructions that went well beyond the balance. Ultimately it would allow him to construct the parabola as the curve describing projectiles.

A more general, and familiar, form of the model is stated later in Chapter 14, which has the title: “… a discussion of the ratios of the [speeds of] motions of the same body moving over various inclined planes.”²⁴ Here Galileo wrote “… a heavy body tends to move downward with as much force as necessary to lift it up, i.e. it tends to move downward with the same force with which it resists rising.”²⁵ This was his prelude to considering the force necessary to draw a given weight up an inclined plane, which, again, he immediately turns into a balance problem about the forces exerted by the weights of bodies depending upon the distances they are from a balance point. Galileo was causally tying together the concepts of weight, distance, force, and balance points (or equilibrium points) to solve his problems about the motions of bodies. He applied this model to bodies on inclined planes and pendulums.

Thus did Galileo, in De Motu, first lay out a model for solving all problems of motion. He argued that the problems of floating bodies (with which he started his text) could all be reduced to problems of the Archimedian balance. He went on to show that all simple machines (the lever, the inclined plane and the pendulum) could be also reduced to balance problems. Free fall of bodies came to be an instance of floating bodies or a balance that had no weight on one side.

In a later work, Le Meccaniche (On Mechanics)26 (composed in 1600), he identifies the properties of all mechanical instruments with the motions of heavy bodies.²⁷ Galileo explicitly uses the concepts of the center of gravity (or center of proportional heaviness or weight) and moment to talk about the simple machines. “Thus, moment [momento] is that impetus to go downward composed of heaviness, position and of anything else by which this tendency may be caused.”²⁸ The model by which such concepts were instantiated and illustrated again was the equilibrium balance model, balancing of the arms or weights on a steelyard – a single-arm balance or lever. Note also that moment (momento) is essentially a generalized force concept. Using this model, Galileo went on to explain the lever, the windlass, the capstan, the pulley, and the screw, and finally he made a first attempt at handling the force of percussion (or impact). These discussions are clearly within the Archimedian tradition and the tradition of the pseudo-Aristotelian mechanics. But it is the reduction of all the Archimedian simple machines and the problems of natural and unnatural motion to the problem of the balance to which I draw your attention.

Galileo used this equilibrium model all of his life. Prior to De Motu he had written The Little Balance (La Balancitta) in 1586.²⁹ In 1612 he wrote a Discourse on Floating Bodies (Discorso Intorno alle Cose che Stanno in su’ l'Acqua o che in Quella si Muovono), which considered hydrostatic phenomena using a causal, equilibrium model.³⁰ The concern with water and its equilibrium came up again in his theory of the tides (Discorso del Flusso e Reflusso del Mare) first in 1616³¹ and then in his Dialogue on the Two Chief World Systems (Dialogo sopra i Due Massimi Sistemi del Mondo Tolemaico e Copernico) in 1632.³²

This balance model takes on larger epistemological force when, in 1623, it becomes the whole of the image in which he sets Lothario

Sarsi's (Orazio Grassi) claims about the comets in The Assayer (Il Saggiatore)33, which he names in contrast to Grassi's 1619 tract, The Astronomical and Philosophical Balance (Libra Astronomica ac Philosophica).³⁴ There Galileo is contrasting ironically many meanings of “balance,” from the balance scale (lances) to the alchemist's fire, to the true tester (saggiatore), to wisdom, to the very notion of justice herself, Libra.

In Dialogo (of 1632)³⁵ it was the model of the balance that provided Galileo the means by which to think about motion and the relativity of perceived motion. So in Day 2 we read:

Salviati: Do you not believe that the tendency of heavy bodies to move downward, for example, is equal to their resistance to being driven upward?

Sagredo: I believe it is exactly so, and its for this reason that two equal weights in a balance are seen to remain steady and in equilibrium.³⁶

In the next passage he goes on to introduce the steelyard, as a way of understanding.

Earlier in Dialogo the balance was again his metaphor for clear thought.

So let us hear the rest of the arguments favorable to his [Aristotle's] opinion so that we may proceed with their testing, refining them in the crucible and weighing them in the assayer's balance [ponderandole con bilancia del saggiatore].³⁷

But balance and equilibrium also mean proper proportionality or right ratio (reason). Human understanding or reason (ragione) is having the correct measure (ratio) for things.

Sagredo: Please, Salviati let us waste no more time invoking these ratios against people who are ready to accept the most disproportionate things…³⁸

Finally in his last work, the 1638 Discourses and Mathematical Demonstrations Concerning Two New Sciences (Discorsi e Dimonstrazioni Mathematiche Intorno a Due Nuove Scienze)³⁹ Galileo used the same concepts of impetus, moment, and center of gravity to solve motion problems about inclined planes, pendulua, free fall, and projectiles (and their parabolic curves.) In Discorsi his model for thinking about the world was the same. He dealt directly with natural motion mostly using the model of the inclined plane, “I assume that the degrees of speed acquired by the same movable over different inclinations of planes are equal whenever the heights of those planes are equal,”40 and then immediately devised an experiment that used a pendulum to prove his point about inclined planes. From there he moved directly to Proposition I, Theorem I about falling bodies (free fall): The time in which a certain space is traversed by a movable in uniformly accelerated movement from rest is equal to the time in which the same space would be traversed by the same movable carried in uniform motion whose degree of speed is one half the maximum and final degree of speed of the previous, uniformly accelerated motion.⁴¹ And from there, he proceeds to his famous result (Prop. II, Theorem II): If a movable descends from rest in uniformly accelerated motion, the spaces run through in any times whatever are to each other as the duplicate ratio of their times; that is, they are as the squares of those times.⁴² All these solutions to problems were called “mechanical conclusions”.⁴³ The inclined plane was again explained by using the balance and talking about equilibrium of weights.

It is of historical interest that this equilibrium model based on the balance and extended to the other simple machines, and using proportional geometry, remained the model for understanding motion problems through the seventeenth century. Descartes's collision models, Huygens’ collision models and his general laws of motion, the work on the laws of motion by Wren and Wallis, and Hooke's work on the spring all used equilibrium as their fundamental problem-solving concept. Even Newton in Principia Mathematica, though he laid the ground for a change away from this relational, geometrical way of doing physics, relied on the Galilean form of proportional balance.

The treatment of all motion by means of Archimedian simple machines gives new insight as to why in this age nature was thought of as machinelike and why some seventeenth-century natural philosophers came to call themselves the mechanical philosophers. Mechanics was the theory of simple machines and so the term “mechanics” came to stand for a part of what now we call physics. But the practitioners of mechanics were artisan craftsmen, who, in contrast to the philosophers and mathematicians, were called “mechanics.” So Simplicio in Dialogo contrasted Aristotle's approach:

… for the accelerated motions he [Aristotle] was content to supply the causes of accelerations, leaving to mechanics and other low artisans the investigation of the ratios of such accelerations and the other more detailed features.”44

The balance as the model for what is intelligible during this period in history had a great and convoluted history. The balance was physically and metaphorically the model of intelligibility of the age. It was obviously observable when the balance was in equilibrium, when the weights and arms were equalized. Any individual could judge when a problem had been solved, when “things were right.” And it was a concept of correctness or proof that could be easily taught. It was a way of interpreting phenomena that anybody could learn, and the standard for success was patent. There was no question of whether you had a proof or not; it was easily seen. Those who would not accept this model of intelligibility would not open their eyes. Personal ambition (such as claiming priority over Galileo) or dogmatism or authority blinded them.

In the late sixteenth century, with rising capitalism and the introduction of the concept of the nation state, the balance become the popular model for bourgeoisie bookkeeping with its “balanced accounts,” for international commerce and its “balance of trade, “⁴⁵ and for the relation between nations with “balance of payments.” Truly the model of Archimedes had permeated the whole of the fabric of society and social relations.⁴⁶ Later, with the idea of the contract as an equilibrium among individuals established by mutual agreement, social relations will come to have a new footing, and government itself will have a new legitimation. The legitimation is taken from the mathematics of mechanics, but this science itself took it from the mathematics and the ideas of commerce and trade for which that math was first used.

GALILEO'S MATHEMATICS

Galileo and other seventeenth-century practitioners of mathematics have been done ill by their anachronistic commentators. Never is it acceptable to put Galileo's proofs and theorems into algebraic form. To do so destroys the mind set, the schema, the very model of intelligibility with which they were working.

For Galileo, mathematics meant geometry. This is the way in which his famous claim in Il Saggiatore must be read:

Philosophy is written in this grand book – I mean the universe – which stands continuously open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these one is wandering about in a dark labyrinth.47

Galileo used a comparative, relativized geometry of ratios as the language of proof and mechanics, which was the language in which the book of nature was written. This is very different from what will follow in the eighteenth century and from the way we think of science today. In very few places in his work, and then mostly in talking about astronomical distances, does Galileo attempt to ascertain real values for any physical constant. Nowhere does Galileo attempt to find out, for example, what the real speed or weight of anything is. This proportional geometry is inherently comparative and relational, a matter of ratios. It measures one thing by showing its relation to another, which then may be quantitatively compared by supplying some arbitrarily or conveniently intelligible standard.⁴⁸ In this sort of geometry there are no absolute values, numbers that describe the true properties of things and so might serve as the touchstone for certainty or objectivity. Using this geometry one does not look for physical constants or solutions to problems in terms of absolute numerical values.⁴⁹

Galileo was not interested in determination of specific numerical properties. Yet Baliani seems to agree and to understand part of Galileo when he writes to Galileo, commenting on his Discorsi to him:

I also think of reasoning about it thoroughly in a treatise I intend to publish on logic, and [there] show that science does nothing but to seek the causes [which] belong to a different habit called wisdom and just as the principles of the sciences are customarily definitions, axioms, and postulates, in physical things these are for the most part experiences, on which are founded astronomy, music, mechanics, optics, and all the rest. (July 20, 1639)⁵⁰

The point is that experiences are the experiences, literally, of seeing mechanical, optical, and astronomical objects as the idealized objects of geometry. All experiences involve seeing things as they are accordingly to your model of intelligibility.

Galileo's geometry is the geometry of inscribed and proscribed circles. It is the geometry of mean proportionals (media proportionales). These are the ways Galileo brought the mixed science tradition to describe the balance. They are also the ways in which Galileo finds his chords theorem of descent (all bodies fall through the chord of a vertical circle in the same time-equal weight or not), and this gave him the times squared law for distances (in his 1602 letter to del Monte).51 The relation between this ratio-geometry and the balance comes out clearly in Discorsi, Theorem IX, where he proves the chords theorem using the model of the balance.⁵² Here is just a little part of the proof, to give a flavor of what Galileo's method is like:

Proposition IX. Theorem IX

If any two planes are inclined from a point in a horizontal line, and are cut by a line that makes with them angles alternately equal to their angles with the horizontal, the movements in the parts cut off by the said line are made in equal times.

From point C of horizontal line X let there be any two inclined planes CD and CE. At any point in line CD construct angle CDF equal to angle XCE; line DF cuts plane CE at F so that angles CDF and CFD equal angles XCE and LCD, taken alternately. I say that the times of descent through CD and CF are equal. Thus, the ratio of the heights of equal planes CD and CE is the same as the ratio of lengths DC and CF. Therefore, the times of descents in these will be equal; which was to be proved.⁵³

This is just the geometry of co-alternate angles and similar triangles, and similar triangles are the balance.⁵⁴ But there is an important addition: These lines can move, just as the balance beam can move. There is a dynamic aspect to this mathematical mechanics. Galileo's commitment to ratios and proportionality as the true method is shown again by the text of the work he was dictating when he died: “On Euclid's Definitions of Ratios.”⁵⁵

This proportional geometry also made it easy to think in terms of relative motion. It is only when one weight is heavier than another that the balance moves. When forces are equal there is no observable motion, though there is force acting. In Galilean relative motion the observer judges what is moving from his point of view, from his reference frame, neglecting all the motions that the observer and the objects have in common. So it is that since a person and all other earthly things, for instance, a boat, a cannon, or a bird, share the same circular motion, that person can only perceive motions that are added to the common motion.

Yet in this person-relative way of looking at the world and judging when proofs had been successful, Galileo recognized a problem. The balance model of intelligibility and the equilibrium model of proof demanded subjectivity. Any one person could look at the balance, or even with many together, each could look, and judge. But this was insufficient and objectivity was needed.

In Day 1 of Dialogo, Galileo contrasted God's extensive knowledge with the human being's intensive knowledge.56 When a person posed a puzzle in the language of proportional geometry where solutions were recognized by seeing that equilibrium was achieved, then the person was Godlike in insight and understanding of the case at hand. The universal can be seen in the particular. By contrast, God sees all of the infinite cases. God's extensive knowledge of all cases only contrasted with human certainty in its intensive, particular mode of operation. The individual could be assured of his certainty intensively by using a proper method of proof, a mechanical, mathematical method.

It was in this way that the more geometrico provided the model of intelligibility and proof for science. The geometry involved was not a pure geometry but the physical geometry of the mixed sciences. It was the geometry of Archimedes, the geometry of proportions and of the properties of machines considered relative (or in relation to) one another. The visual paradigm of equilibrium proof for the balance brought together the Galilean tenets of experiment, long observation, and rigorous demonstration.

Indeed, this proportional geometry and its attendant equilibrium balance was the model for all natural science until Newton changed the ground rules with algebra replacing proportional geometry, absolute space replacing relational place, true motion replacing relative motion, and God becoming an active intervener in the world in demonstrable ways and equilibrium. The balance as a model of intelligibility in physics gave way to algebraic equality; understanding the world by relating one thing to another in human terms became a problem of solving the equation in order to find a real number value for a given force. These changes gave science a new model of intelligibility and success.

GALILEO'S EXPERIMENTS AND EXPERIENCES

Experiments for Galileo were ways of providing for himself and others first-person experiences in order to discover or verify relationships holding in nature. They were ways of demonstrating that the phenomenon that had been geometrically described actually could produce the results claimed. These experiences made the terms of the natural mechanical explanations meaningful and real to the person constructing or discovering.

Mechanical models, comparable production of effects, and commonplace experiences all played the same set of roles in Galileo's thought. They were not merely rhetorical devices invoked to convince others about the certainty of conclusions, they were also epistemic requirements necessary to give the proper experience to the individual scientist putting forward the explanation. Thus in Discorsi Galileo has Salviati reply to Simplicio's query about falling bodies as to whether “this is the acceleration employed by nature in the motion of her falling heavy bodies.”

Like a true scientist, you make a very reasonable demand, for this is usual and necessary in those sciences which apply mathematical demonstrations to physical conclusions, as may be seen among writers on optics, astronomers, mechanics, musicians and others who confirm their principles with sensory experiences that are the foundations of all the resulting structures. Therefore as to experiments, the Author [Galileo} has not failed to make them, and in order to be assured that the acceleration of heavy bodies falling naturally does follow the ratio expounded above, I have often made the following test [la prova] in the following manner.57

“Prova” here does mean test, but it is used similarly to “proof” in the phrase, “the proof (or test) of the pudding is in the eating.” The proof lies literally in showing what happens. Even in his letter to Castelli in 1613, Galileo speaks of “those natural conclusions of which the manifest meaning or the necessary demonstration have made certain and sure.”⁵⁸

From the time of De Motu onwards (1590), Galileo held that the way to understand anything was to show how it worked in a mechanical way. As noted, this probably derived from his practical training in the mechanical arts, his so-called artisan-engineering background. This was also a reasonably new Renaissance enterprise and was quite individualistic in that training was given outside of traditional guilds and schools in an apprenticelike or tutorial fashion.

It was a practical do-it-yourself procedure. It was practical in that the practitioners designed instruments, built fortifications, and constructed machines. These were useful devices. But it was practical in that it involved practices. One could understand how these machines worked by constructing them and seeing what they did. For big projects one built models. But whether they worked or not was tested by the very practice of constructing them and seeing if they worked.

Intelligibility or having a true explanation for Galileo had to include having a mechanical model or representation of the phenomenon. In this sense, Galileo added something to the traditional criteria of mathematical description (from the mixed sciences) and observation (from astronomy) for constructing scientific objects (as some would say) or for having adequate explanations of the phenomena observed (as I would say). He puts it nicely in the negative, when criticizing alternative theories of the tide. But the moral is clear: To get at the true cause, you must replicate or reproduce the effects by constructing an artificial device, so that the effects can be seen.

I believe you do not have any stronger indication that the true cause of the tides is one of those incomprehensibles than the mere fact that among all the things so far adduced as vera causa there is not one that we can duplicate for ourselves by means of appropriate artificial devices.59

Galileo did draw a diagram and construct a mechanical device to prove that the Earth's motion was the true mechanical cause of the flux and re-flux of the tides. Diagrams are useful but insufficient. In talking again about the tides and Copernicus's third motion of the Earth Salviati explains to Sagredo:

… Nevertheless, we shall see whether drawing the drawing of a little diagram will not shed some light on it. [However] It would be better to represent this effect by means of solid bodies than a mere picture.⁶⁰

For Galileo to have an explanation he had to have suitable experiences demonstrating that the explanatory cause is the true one and that it works necessarily. These experiences are had by constructing machines that duplicate the phenomenon in question and so demonstrate or make plain their workings. Where one cannot construct a machine, one may use examples already constructed or found in nature that exhibit or display how the phenomenon works. These devices or analogic phenomena literally demonstrate how the phenomena occurs. One sees the machine or phenomenon as an instance of mathematics in the world. In this way one can know that they are real and not imaginary.

This idea of mechanical models or real or constructed experiences being needed for demonstration fits well with the constructive geometrical tradition. To have a demonstration in geometry one must construct a proof, actually draw the diagram. This idea of active construction is also the force behind the persuasiveness of the passage in Dialogo where Salviati has Simplicio construct the diagram, using agreed upon experiences as constraints, that shows the relations among the planets.61 In this way, Salviati says he will come to understand even though he does not believe. Of course, Simplicio ends up with a Copernican diagram. To have a demonstration in mechanics one must construct a geometrical proof and coordinate it with experiences.

For motion problems this meant the model of intelligibility was one of the Archimedian simple machines, first the balance and then later the inclined plane and pendulua. In the theory of the tides it was a constructed wheel with an embedded tube of water that could be rotated. In the Letters on the Sunspots it was throwing bitumen on a hot pan thereby creating clouds of smoke that were like the spots on the sun.⁶² These are all models. These mechanical models were a necessary part of the “proof” or criteria of adequacy for determining if you had an explanation. What is mechanical in this sense can be drawn or reproduced in a picture or recipe book. Such things can be seen or made by everyone and anyone.

I would remind you of the extent to which little machines, constructions, and natural phenomena are used by Galileo as demonstrations or parts of arguments. I will provide a few interesting examples, but literally a hundred or so more fill the pages of this book. Just in Dialogo we have: pouring water on the pavement to demonstrate reflective properties of the Moon “to show this to your own senses,”⁶³ using a pendulum to represent motions,⁶⁴ observing flying animals while you are on a moving ship,⁶⁵ using steelyards and balances,⁶⁶ using a “material instrument” – an astronomical sphere – for representing facts about the Sun's rotation,⁶⁷ solving a ratio problem about the size of stars with a real piece of rope that blocks out the adventitious rays of the stars,⁶⁸ and arguing about the time of rotation of a body (the Earth) by showing how to regulate the time in wheel clocks.⁶⁹ Finally, Galileo feels the need to tackle the toughest problem for all of the mechanical philosophy and for any form of corpuscularianism, even atomism, when in Dialogo, he attempts to provide a mechanical explanation of Gilbert's magnetism in terms of smooth and touching particles that fill the holes of a sponge.70

PHILOSOPHICAL CONCLUSION: THE NATURE OF MODELS OF INTELLIGIBILITY⁷¹

I want to call Galileo's use of the balance, conceptual and real, a model of intelligibility. It is the model by which he actually does his science, and it directs what he looks for and how he thinks about the world. It also includes criteria for making it clear when he has succeeded in explaining something.

Talking about a model of intelligibility is not intended as a new way of speaking. It is similar in some aspects to many other concepts that have been used before. But the emphasis on its being a model (or representation) that directs thought, and the concomitant understanding, is stressed more by this locution. For example, Stephen Toulmin's ideals of natural order,⁷² Stephen Peppers’ world hypotheses,⁷³ and Gerald Holton's thematics⁷⁴ all were similar attempts to talk about ways of making sense of the world. But these seem too abstract, too diffuse, and too weltanschaung-like to capture the precision with which Galileo's model of intelligibility directs his work.

The mechanical world view, which is a favorite example of these writers, would not have been either possible or plausible without the intelligibility and form of understanding provided by Archimedian simple machines, especially the balance. Its physical concreteness, mathematical describability, and physical manipulability leading to experimental possibilities gave intelligibility and structure to the abstract concepts of the mechanical world picture.

Closer to my concept of a model of intelligibility is Thomas Kuhn's concept of an exemplar.⁷⁵ For Kuhn, an exemplar is a problem-solving schema that is learned through using it on problems. It is based on shared examples. Certainly this is a model of intelligibility. However, Kuhn sometimes seems to think of exemplars as rules of thumb used to solve algebraic story problems or, sometimes, abstract algorithms or mathematical schemas that are learned by use and then applied in similar kinds of cases. These lack an intuitive picturable intelligibility. Often they are too ad hoc, and seldom do they have a broad enough domain of application. In other cases, he seems to think they are generalized examples, and in some sense the balance and solving problems by reducing them to balance problems are examples, but these now are too general to fit the concrete idea of an exemplar that Kuhn seems to have in mind.

The picturable, perceptual character of our model of intelligibility is caught in some ways like some of Eleanor Rosch's perceptual prototypes.76 A model of intelligibility does provide a meaning schema by directing attention to what is important in a problem and by exhibiting what relations exist among those important elements. The model directs attention to the elements or parts of the problem that need to be identified and selected, making the subject match one element on the model with one element in the problem.

However, there is no algorithm for adequate or correct mapping. Misidentification of the elements can occur. Also, parts of the problem may not have correlates in the model either because they have been suppressed (in which case they need to be supplied) or because the problem is not really of the type that can be solved by the model. In this latter case again, there is no certain way to tell when a problem solution by the model is possible. Further, prototypes of a type of phenomenon may all exhibit certain “accidental” features which must be ignored in order to find them intelligible. This is why Galileo insists always that science is based on perception and reason. Rosch's prototypes are too exclusively perceptual for my purposes.

Further, all models of intelligibility need not have a picturable, perceptual character. In fact, a model of intelligibility can be very abstract and highly conceptual. Maxwell's electromagnetic equations or the U.S. Constitution are examples.

In its cognitive aspects, a model of intelligibility resembles some theorists’ descriptions of the workings of schemas,⁷⁷ plans,⁷⁸ theories or hypotheses,⁷⁹ mental models,⁸⁰ problem-solving processes,⁸¹ or idealized cognitive models.⁸² Models of intelligibility are rules for drawing inferences. The models also specify patterns of expectation, group certain phenomena or situations into classes, and constitute the structures or categories into which the world must fit for us to find it comprehensible and intelligible.

However, models of intelligibility depart from this psychological class of concepts by existing, in some sense, apart from the person. They are not just abstractions but can be instantiated in concrete objects and are part of social or cultural conditions. Their use in thinking is as idealized objects; otherwise they could have no normative force. But their intelligibility and utilization derives much from the fact that often they are spatially representable and thus picturable. Further, their physical presence or representation allows actions to be performed on them, and relations among their parts can be literally discovered and seen. Thus, they lend themselves to experimental and observational possibilities. This also explains why they can be used to train students.

Part of this physical sense of the model and its independence of human beings was seen by J. J. Gibson83 when he developed his realistic theory of perceptual invariants. These were physical, higher order relational properties of the environment that determined what people perceive.⁸⁴ However, Gibson made a firm distinction between perception and cognition, which is untenable at the level at which models of intelligibility function. They are perceptual and cognitive.

There are interesting parallels, too, between a model of intelligibility and stages of Piaget's developmental psychology, wherein a child moves from sensori-motor structures to concrete operations and then to the more abstract sphere of formal operations.⁸⁵ Even though a model is representable in the real world, it is not merely a concrete, real-world token. It also is an ideal psychological type that can be taught through its concrete exemplifications and through physical representations (drawings, diagrams, etc.). Because it is ideal, its structure can function to regulate and constrain applications or instantiations of itself. That is, its ideal character allows one to correct representations or applications of it by comparisons back to the ideal model. The model can thus function normatively as well as descriptively.

A model of intelligibility's structure can be accepted as understood or intelligible. A model of intelligibility can be extended and used to understand other types of cases, bringing unity to different domains and fields.

Finally, the intelligibility of a model of intelligibility allows it to help bring about agreement among people. Because a model exhibits all and only those properties that are important, people can check over the list of properties. This intelligibility and the normative character of the idealized model are what allows for objectivity. If a problem cannot be reduced to these elements, or if a participant in the investigation insists on attending to other aspects, then either the problem falls outside the scope of the model or the participant needs (re)training about what is important in the problem or what are the allowable procedures.86 Because these are the factors that most often lead to disagreements and disputes, such disagreements can be used to test the scope and adequacy of models, and sometimes they give rise to “revolutions” in intelligibility when people become convinced that something important is being left out. This latter may cause them to abandon the model as inadequate for solving all problems or merely to establish it as limited in application to certain kinds of cases or problems.

This is what happened to the Galilean mechanical model of intelligibility after Newton changed the terms and the methods and began to change the mathematics of science.

NOTES

  1 I offer great thanks to all those who have helped me with Galileo over the years. There are too many to name, but to all of you, I owe a great deal. I am sorry that most of you will not agree with this essay.

         Many of the Galileo references are to Antonio Favaro's Le Opere di Galileo Galilei, Edizione Nazionale, Firenze; reprinted in 1968. References to works in this edition will simply be listed by author, then EN, followed by volume number, and page number.

  2 Galileo (Rome 1623), The Assayer, in S. Drake and C. D. O'Malley, The Controversy of the Comets of 1618, Philadelphia: University of Pennsylvania Press, 1960, p. 189; Il Saggiatore, Milano: Feltrinelli, 1992, p. 48.

  3 For some more detail see Peter Machamer, “The Person Centered Rhetoric of Seventeenth Century Science,” in Marcello Pera and William R. Shea, eds., Persuading Science: The Art of Scientific Rhetoric, Canton, MA: Science History Publications, 1991.

  4 There were very many practitioners of the mechanical arts before Galileo, and they were becoming popular and intellectually recognized. Some immediate predecessors are collected in Stillman Drake and I. E. Drabkin, Mechanics in Sixteenth Century Italy, Madison, WI: The University of Wisconsin Press, 1969. But such arts predate the sixteenth century and even printing, see, e.g., Pamela Long, “Power, Patronage, and the Authorship of Ars: From Mechanical Know-How to Mechanical Knowledge in the Last Scribal Age,” Isis, 88, 1997, pp. 1–41. There is much work to be done on this history of early machine theory and mechanics and its dissemination, and especially in its relation to printed, widely available texts. Printing allowed mechanical diagrams for teaching, which was a radical new representation of knowledge.

  5 See, for example, Otto Mayr, Authority, Liberty and Automatic Machinery in Early Modern Europe, Baltimore, MD: The Johns Hopkins University Press, 1986.

  6 There were many participants in these method debates of Galileo, but the interested reader might look at Alexandre Koyré, From the Closed World to the Infinite Universe, Baltimore, MD: The Johns Hopkins University Press, 1957; Ludovico Geymonat, Galileo Galilei: A Biography and Inquiry into His Philosophy of Science, translated by Stillman Drake, New York: McGraw-Hill 1965; Ernan McMullin, “Introduction: Galileo, Man of Science,” in Ernan McMullin, ed., Galileo: Man of Science, New York: Basic Books, 1967, pp. 3–51; Dudley Shapere, Galileo: A Philosophical Study, Chicago, IL: The University of Chicago Press, 1974; Thomas P. McTighe, “Galileo's Platonism: A Reconsideration,” in Ernan McMullin, ed., Galileo: Man of Science, op. cit., pp. 365–87; William R. Shea, Galileo's Intellectual Revolution, New York: Science History Publications, 1972.

  7 See Footnote 4.

  8 Ernst Cassirer, The Individual and the Cosmos in Renaissance Philosophy, translated by Mario Domandi, Philadelphia, PA: University of Pennsylvania Press, 1972 (original 1927).

  9 Olaf Pedersen, The Book of Nature, Vatican City: Vatican Observatory Publications, 1992.

10 Interestingly E. J. Dijksterhuis, though author of books of the mechanical world view and on Archimedes, misses the depth of this connection; see The Mechanization of the World Picture, translated by C. Dikshoorn, London: Oxford University Press, 1961 (original 1950).

11 See, for example, Leonardo Olschki, Galileo und seine Zeit, Halle, 1927 and “Galileo's Philosophy of Science,” Philosophical Review 1943, pp. 349–65; Erwin Panofsky, Galileo as a Critic of the Arts, Nijhoff, 1954; Tom Settle, “Galileo's Use of Experiment as a Tool of Investigation,” in E. McMullin, ed., Galileo: Man of Science, New York: Basic Books, 1967.

12 Wallace Hooper, Galileo and the Problems of Motion, Dissertation, Indiana University, 1992. Hooper's thesis is one of the best works on Galileo and motion that exists.

13 Peter Machamer “Galileo and the Causes,” in Robert Butts and Joseph Pitt, eds., New Perspectives on Galileo, Dordrecht: D. Reidel, 1978, pp. 161–80. Cf. also James G. Lennox, “Aristotle, Galileo and the ‘Mixed Sciences’,” in William Wallace, ed., Reinterpreting Galileo, Washington, DC: The Catholic University of America Press, 1986.

14 William Wallace, Galileo's Logic and Discovery and Proof, Dordrecht: Kluwer 1992.

15 Galileo, De Motu, 1590, EN vol. 1, pp. 251–419; translation by I. E. Drabkin, On Motion, Madison WI: The University of Wisconsin Press, 1960. Hereafter, this translation is referred to as “Drabkin” followed by page number. As to the dating, see review of attempts by Wallace Hooper, op. cit.

16 See Wallace Hooper, op. cit., p. 255 f.

17 Galileo, EN 1, 174; Drabkin 38.

18 Galileo, EN 1, 259; Drabkin 22.

19 Galileo EN 1, 259; Drabkin 23.

20 Galileo, EN 1, 257; Drabkin 20.

21 Galileo, EN 1, 257–8; Drabkin 20–21.

22 Galileo, EN 1, 259; Drabkin 22–23.

23 Galileo, EN 1, 274, 275; Drabkin 38, 39.

24 Galileo, EN 1, 296; Drabkin 63.

25 Galileo, EN 1, 297; Drabkin 64.

26 Galileo, EN 1; translated by Stillman Drake, On Mechanics, Madison, WI: The University of Wisconsin Press, 1960.

27 Galileo, EN 1; Drake 151.

28 Galileo, EN 1; Drake 151, my translation.

29 Galileo, EN 1, 215–16; translation in Laura Fermi and Gilberto Bernardini, Galileo and the Scientific Revolution, Greenwich, CT: Fawcett, 1961.

30 Galileo, EN 4; Stillman Drake translated this work and made it into a dialogue by adding his own thoughts, Cause, Experiment and Science, Chicago: The University of Chicago Press, 1981.

31 Galileo, EN 5, 378 ff.

32 Galileo, EN 7; translation by Stillman Drake, Dialogue on the Two Chief World Systems, Berkeley and Los Angeles: The University of California Press, 1962. Hereafter this work will be referred to as Dialogo.

33 Galileo EN 6; translation by Stillman Drake, The Assayer in Galileo, et al.; Controversy of the Comets of 1618, Philadelphia: The University of Pennsylvania Press, 1960.

34 Grassi, EN 6; translation by C. D. O'Malley, in Galileo et al. Controversy of the Comets of 1618, op. cit.

35 See Footnote 24.

36 Galileo, EN 7, 240; Drake 213–14.

37 Galileo, EN 7, 157; Drake 131.

38 Galileo, EN 7, 393–4; Drake 366.

39 Galileo, EN 8; translation by Stillman Drake, Two New Sciences, Madison, WI: The University of Wisconsin Press, 1974. Hereafter, this work is referred to as Discorsi.

40 Galileo, EN 7, 205; Drake 131.

41 Galileo, EN 8, 208; Drake 366.

42 Galileo, EN 8, 209; Drake 166.

43 Galileo, EN 8, 214; Drake 171.

44 Galileo, EN 7, 190; Drake 164.

45 W. H. Price, “Origins of the Phrase ‘Balance of Trade’,” Quarterly Journal of Economics, 1905, pp. 157–67.

46 For a review of some of this, see Philip Mirowski, More Heat than Light: Economics as Social Physics, Physics as Nature's Economics, Cambridge: Cambridge University Press, 1989.

47 Galileo Galilei, Il Saggiatore, 1623, Milano: Feltrinelli, 1965, p. 38; Drake translation in Controversies of the Comets of 1618, op cit. pp. 183–4.

48 I have struggled to find a way of describing Galileo's form of mathematico-physical reasoning. The ideas of ratio, proportionality, congruence, and dynamic motion are all part of it, but I still have no felicitous way of describing its character succinctly or precisely.

49 Stillman Drake understates this proportional character of Galileo's geometry when he writes:

     Unlike Galileo who concerned himself only with ratios, not only Baliani, but other scientists of his time and after (Mersenne, Cabeo, G. B. Ricioli, and the young Newton) sought number of feet or other arbitrary units traversed in fall from rest during a given number of astronomical seconds. Some figures given in the Dialogo as an example for purposes of calculation (EN 7, 50, Dialogue p. 223) were accordingly misunderstood as assertions of an experimental determination.

     Stillman Drake, Galileo at Work, Chicago: The University of Chicago Press, Footnote 7, p. 498.

50 Quoted in Drake, Galileo at Work, p. 398. Drake, of course, assumes Baliani is disagreeing with Galileo.

51 This development is brilliantly laid out in Wallace Hooper's Galileo and the Problems of Motion, op. cit., p. 344 ff.

52 Galileo, EN 8, 227–8; Drake 184–5.

53 Galileo, EN 8, 227–8; Drake 184.

54 Wallace Hooper, Galileo and the Problems of Motion, op. cit., p. 348.

55 Translated in Stillman Drake, Galileo at Work, op. cit., pp. 422–36.

56 Galileo, EN 7, 128–9; Drake 103.

57 Galileo, EN 8, 212; Drake 169.

58 Galileo, EN V, 279–88.

59 Galileo, EN 7, 447; Drake 421.

60 Galileo, EN 7, 482; Drake 457. Unfortunately, even Galileo's artificial machine did not prevent him from accepting a wrong theory of the tides. But to his credit, lunar gravitation is a “miraculous force” and cannot be fit into the mechanical philosophy.

61 Galileo, EN 7, 350; Drake 322. But for your greater satisfaction and your astonishment, too, I want you to draw it yourself. You will see that however firmly you may believe yourself not to understand it, you do so perfectly and just by answering my questions you will describe it exactly.

62 For discussion of this technique and all the relevant quotations, see Rivka Feldhay, “Producing Sunspots on an Iron Pan: Galileo's Scientific Discourse” and my reply, both in Henry Krips, J. E. McGuire, and Trevor Melia, eds., Science, Reason and Rhetoric, University of Pittsburgh Press/Unveritatsverlag Konstanz, 1995, pp. 119–43, 145–52.

63 Galileo, EN 7, 123; Drake 97–9.

64 Galileo, EN 7, 177–8; Drake 152–3: “The pendulums have just shown us that the less a moving body partakes of weight, the less apt it is to conserve motion …” and EN 7, 454; Drake 428, for a swinging stone or pendulum representing the reciprocal motion of the tides.

65 Galileo, EN 7, 212–3; Drake 186–7.

     For a final indication of the nullity of the experiments brought forth, this seems to me the place to show you as a way to test [il modo di sperimentarle] them all very easily. Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals…

66 Galileo, EN 7, 240–1; Drake 213–4.

     Sagredo: But tell me what is this second force.

     Salviati: It is that which did not exist in the equal armed balance. Consider what there is that is new in the steelyard, and therein lies necessarily the cause of the new effect.

67 Galileo, EN 7, 375–6; Drake 348–9.

68 Galileo, EN 7, 388–9; Drake 361–2.

69 Galileo, EN 7, 474–5; Drake 449–50.

70 Galileo, EN 7, 435–6; Drake 409–10.

71 A version of this section appeared in Peter Machamer and Andrea Woody, “A Model of Intelligibility in Science: Using Galileo's Balance as a Model for Understanding the Motion of Bodies,” Science and Education 3, 1994, pp. 215–44.

72 Stephen Toulmin, Foresight and Understanding, New York: Harper, 1961.

73 Stephen Pepper, World Hypotheses, Berkeley, CA: The University of California Press, 1948.

74 Gerald Holton, Thematic Origins of Scientific Thought: Kepler to Einstein, Cambridge, MA: Harvard University Press, 1972.

75 Thomas Kuhn, The Structure of Scientific Revolutions, second edition, Chicago, IL: The University of Chicago Press, 1970, “Postscript.”

76 For example, Eleanor Rosch, “Cognitive Reference Points,” Cognitive Psychology, 1975 7:532–47.

77 Ulrich Neisser, Cognition and Reality, San Francisco, CA: W. H. Freeman, 1976; Julian Hochberg, Perception, second edition, Englewood Cliffs, NJ: Prentice-Hall, 1978.

78 George A. Miller, E. Galanter, and K. Pribram, Plans and the Structure of Behavior, New York: Holt, Rinehart, and Winston, 1960.

79 Richard L. Gregory, “Choosing a Paradigm for Perception,” in Edward C. Carterette and Morton P. Friedman, eds., Handbook of Perception, Vol. 1, New York: Academic Press, 1974.

80 Philip Johnson-Laird, Mental Models, Cambridge, MA: Harvard University Press, 1983.

81 For example, Herbert A. Simon, “Thinking by Computers” and “Scientific Discovery and the Psychology of Problem Solving,” both in Robert G. Colodny, ed., Mind and Cosmos, Pittsburgh, PA: University of Pittsburgh Press, 1966, pp. 3–21, 22–40. Another, nonpsychological approach using problem solving as a model for how research traditions are structured is found in Larry Laudan, Progress and Its Problems, Berkeley, CA: University of California Press, 1977.

82 George Lakoff, Women, Fire and Dangerous Things, Chicago, IL: The University of Chicago Press, 1987.

83 James J. Gibson, The Senses Considered as Perceptual Systems, Boston, MA: Houghton Mifflin, 1966, and The Ecological Approach to Visual Perception, Boston, MA: Houghton Mifflin, 1979.

84 A theory, not unlike Gibson's, in some respects, is developed for the case of motion (and so for event perception) in James E. Cutting, Perception with an Eye for Motion, Cambridge, MA: MIT Press, 1986.

85 Piaget has developed this theory in numerous publications. For a summary, see Jean Piaget and Barbel Inhelder, The Psychology of the Child, translated by H. Weaver, New York: Basic Books, 1969 (originally 1966).

86 This theme is developed, albeit in a dated Wittgensteinian language of criteria, in William G. Lycan and Peter Machamer's, “A Theory of Critical Reasons,” in B. R. Tilghman, ed., Language and Aesthetics, Witchita, KS: Kansas State University Press, 1971.