The Meeting with Infinity

Galileo's last book is usually referred to by the simplified title Two New Sciences. This time the three friends, Salviati, Sagredo and Simplicio meet after visiting the Arsenal, Venice's famous shipyard. Salviati is impressed with all the practical knowledge accumulated there:

"The constant activity which you Venetians display in your famous arsenal suggests to the studious mind a large field for investigation, especially that part of the work which involves mechanics; for in this department all types of instruments and machines are constantly being constructed by many artisans, among whom there must be some who, partly by inherited experience and partly by their own observations, have become highly expert and clever in explanation."109

The first of Galileo's "two new sciences" is an attempt at a technical treatment of the characteristics of matter, with special emphasis on fracture and deformation. The book opens with a discussion about the way a large ship is more prone to break up due to its own weight, than a small one built to the same proportions.

Salviati quite correctly states that one can raise a small obelisk without difficulty, whereas a large one of the same proportions is likely to fracture under its own weight.110 This leads on to far more fundamental questions: what in fact holds matter together? How is it built up?

It is remarkable how much Simplicio's role has changed since his appearance in the Dialogue. He is no longer the naive and slightly unsophisticated Aristotelian who invites the sarcastic comments of the others. His function now is to act as an intermediary between Aristotelian physics and the mathematically orientated physics of Galileo. Whenever he introduces Aristotle's observations, they are received with deep respect by his conversational partners. Sagredo even quotes one "infallible maxim of the Philosopher".111 This is a world away from the caustic criticism that Simplicio's hidebound intellectual conservatism unleashes in the Dialogue, where Aristotle's influence is regarded as the greatest bar to scientific progress. But in terms of literature, this sea-change causes the tension between the characters to slacken, making Two New Sciences a much less engaging read.

Salviati's speculative account of the construction of material is based on the traditional understanding (shared by Aristotle) that "nature abhors a vacuum".112 He assumes that all material is composed of "atoms" - the smallest, indivisible entities of the material - which are held together by minute vacuums - vacua - which exert what might be called "negative pressure". This pressure keeps the material firm and intact, but there must be a great many such vacua in the toughest and least breakable materials. Galileo is actually calculating the weight of atmospheric pressure here, without realising it.

Through quite complex geometrical reasonings the disputants arrive at a point where they find themselves almost forced to their knees - before infinity. Simplicio jumps in and protests at the idea that a finite line has an infinite number of points along it - for, as he says, a long line must contain more points than a short one, but it is meaningless to say that one infinite number is higher than another.

Salviati demonstrates, in a most elegant way, that the concepts of "larger" and "smaller" cannot be applied to the infinite.113 He takes numbers as his example. The amount of ordinary numbers is obviously infinite. But every number has a square (22 = 4,32 = 9,42 = 16 etc.). Thus the number of squares is also infinite - even though the sequence is clearly "smaller" because it does not contain the numbers in between the squares.

Galileo - through Salviati - does not stop here, even though he admits that our limited human intellect may not be able to grasp the infinite. He also indicates that there may be something midway between the finite and the infinite, and that there are quantities which can be described using whichever numbers one wants. The number of points on a line is perhaps just one such "halfway house".

In this, Galileo was fairly close to a truth that was first fully revealed 250 years later: in 1874 Georg Cantor proved that there were several classes of infinity. In fact, the points on a line belong to the class that cannot be "be organised" into a progression of numbers. But these speculations show the intellectual force that still lived on in the old prisoner of Il Gioiello, and how he had retained his appetite for approaching the most fundamental problems with reasoning and strict logic.

The continuation of his reasoning is, however, strange and not so easy to understand. Salviati points out that the distance between the squares becomes bigger the greater the square roots become. Therefore this cannot be "the road to infinity", on the contrary it becomes more and more distant as the numbers increase. And so, the only really infinite number is 1!114 It contains every power (12 = 1,13 = 1 etc.)

Galileo was a thorough-going rationalist and mysticism of any sort was foreign to him. But just here it is tempting to think that his mathematics has rubbed up against the boundary of metaphysics. Beyond it lies infinity, which can be summed up in the number one. And who is the One who contains Infinity? Who can it be but God himself.

The three of them press on. Salviati discusses the problem of the speed of light. Does light spread instantaneously, i.e. infinitely fast, or just extremely quickly? He even suggests an experiment to decide the question.115 (The experiment was not precise enough because the speed of light is so great. But in a sense Galileo contributed when the speed of light was measured for the first time, by the Danish astronomer Ole Romer in 1676. The measurement was done using the satellites of Jupiter.)

But this is the end of uncertainty and vagueness. The remainder of the first day is a veritable scientific triumph through the laws of motion, in which all of Galileo's experiments with free fall and pendulums are presented and summed up in exemplary fashion. Simplicio with his Aristotelian counterarguments is amicably and respectfully put in his place. Especially masterly as a piece of scientific prose, is the long section where Salviati argues a proposition that seems quite improbable to Simplicio, namely that a wisp of wool and a lead ball will fall at exactly the same speed in a total vacuum. 116

Salviati provides a careful account of air resistance. He also attempts to calculate the buoyancy of air, clearly based on experiments. The lack of accurate measuring instruments makes his estimate relatively imprecise. Salviati assumes that water is 400 times heavier than air, the correct figure being approximately 780 times. Otherwise his reasoning is so elegant and convincing that Simplicio announces that, if he were about to begin his studies afresh, he would start by reading mathematics!

The first day concludes with a section on pendulums, in which the vitally important law that states that the oscillation time of a pendulum is proportional to the square root of its length, is thrown in almost as an afterthought.117 Much more space is dedicated to a fairly long exegesis about musical theory, which uses his experience of pendulums on swinging strings.

Galileo is here extending and rounding off the work of his father, with precise observations on the relationship of the strings' weight, length and taughtness - and the tones that result. It is all brought to a conclusion by Salviati explaining which intervals sound sweet to the ear and which jarring. The cause is purely mathematical:

"The pulses delivered by the two tones, in the same interval of time, shall be commensurable in number, so as not to keep the ear-drum in perpetual torment, bending in two different directions in order to yield to the ever-discordant impulses."118

Put briefly: the number of vibrations (and thus the tone) must have a harmonic relationship, e.g. 2:3. Everything is a matter of proportion - even musical harmony.

At this point the three take a well deserved rest until the next day. The second day offers a fairly brief and technical description of how one calculates the breaking strength of various bodies. Salviati returns to the starting point of the conversation, and demonstrates geometrically why large constructions are proportionately more vulnerable than small ones -and he explains that is why giants, many times bigger than ordinary people, cannot exist. If they did, their joints at least would have to be made of some other material!119 Yet again Simplicio is there with solid, sensible objections: whales, he says, are monstrously large.120 And so Salviati gets the opportunity to elucidate on the effect of buoyancy; a recurring theme in Galileo's thinking throughout his life.

But it is the third day that is the most important in Two New Sciences. During it, the full panoply of the other new science is presented: the science of motion, kinematics. No longer is Galileo so concerned about maintaining the fiction of the three interlocutors. The chapter opens with a short dissertation in Latin - in his own name. The introduction is like a triumphal fanfare for the work of a lifetime:

"My purpose is to set forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small; nevertheless I have discovered by experiment some properties of it which are worth knowing and which have not hitherto been either observed or demonstrated."121

During the conversation that follows, Simplicio and Sagredo begin to discuss the reasons why objects move. Salviati interrupts them politely but firmly by saying that there are many causes: "the attraction of the centre" (the force of gravity), the influence of the medium they are moving in, a force acting between the basic elements within the object.122 But, he says, Galileo's method is to investigate and show how motion occurs, not why.

Salviati is thus encapsulating something ofwhat is most crucial in Galileo. His mathematical mode of thought provides a description of what actually happens, whereas the traditional Aristotelian logic was always concerned with speculations about cause and effect, without grounding itself in a sufficiently stringent description of reality. Even Simplicio eventually begins to understand a little of this when he extols the exactness of mathematics.

This is the only place in Two New Sciences where the suspicious reader might capture an echo of the debate surrounding the Copernican system. Salviati speaks about what can happen to someone who puts forward irrefutable proof that old, deep-rooted notions are erroneous:

"[There is] a strong desire to maintain old errors, rather than accept newly discovered truths. This desire at times induces them to unite against these truths, although at heart believing in them, merely for the purpose of lowering the esteem in which certain others are held by the unthinking crowd."123

Wisely, Galileo does not continue along these lines. But he clearly believes that this description would fit a Grassi, a Scheiner - and maybe even an Urban VIII - well.

The fourth day also concerns motion - but this time "forced", not "natural" motion (like free fall). The starting point is the sadly practical application such investigations have in ballistics. Musket and cannon balls fly in this chapter. By means of elegant conic section geometry, Galileo (Salviati is now again merely a commentator) proves that, if the ball is fired horizontally, its trajectory is parabolic - provided one accepts his assertion that the ball's curved line of movement can be analysed as consisting of two entirely independent motions. One is the even movement on the horizontal plane imparted by the power from the weapon, the other is the free fall which affects all bodies.

This insight is possibly as important as the law of fall. It forms the basis of all practical descriptions of actual motion.

Salviati promises that the three of them will meet again to talk about impact, that is, brief contact between bodies - on the work's fifth day. But it was never written.

After that Two New Sciences, and Galileo's scientific output, end with the appendix. This comprises a fifty-year old paper about the centre of gravity in bodies. The old prisoner in Il Gioiello delves into the thoughts of the

23-year old who went to Rome and disputed with the Jesuits. And so his life's work is brought to a close.

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