The Discourses and Mathematical Demonstrations Relating to Two New Sciences (Discorsi e dimostrazioni matematiche, intorno à due nuove scienze, 1638) was Galileo's final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years.
After his Dialogue Concerning the Two Chief World Systems, the Roman Inquisition had banned publication of any work by Galileo, including any he might write in the future.[1] After the failure of attempts to publish the work in France, Germany, or Poland, it was picked up by Lodewijk Elzevir in Leiden, The Netherlands, where the writ of the Inquisition was of little account (see House of Elzevir).
The same three men as in the Dialogue carry on the discussion, but they have changed. Simplicio, in particular, is no longer the stubborn and rather dense Aristotelian; to some extent he represents the thinking of Galileo's early years, as Sagredo represents his middle period. Salviati remains the spokesman for Galileo.
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The sciences named in the title are the strength of materials and the motion of objects. Galileo worked on an additional section on the force of percussion, but was not able to complete it to his own satisfaction.
The discussion begins with a demonstration of the reasons that a large structure proportioned in exactly the same way as a smaller one must necessarily be weaker known as the square-cube law. Later in the discussion this principle is applied to the thickness required of the bones of a large animal, possibly the first quantitative result in biology, anticipating J.B.S. Haldane's seminal work On Being the Right Size, and other essays, edited by John Maynard Smith.
Thomas Bradwardine was the first to formulate the equation for the displacement s of a falling object, which starts from rest, under the influence of gravity for a time t (the essential principle had been previously stated by the Oxford Calculators):
In Two New Sciences Galileo (Salviati speaks for him) used a wood molding, "12 cubits long, half a cubit wide and three finger-breadths thick" as a ramp with a straight, smooth, polished groove to study rolling balls ("a hard, smooth and very round bronze ball"). He lined the groove with "parchment, also smooth and polished as possible". He inclined the ramp at various angles, effectively slowing down the acceleration enough so that he could measure the elapsed time. He would let the ball roll a known distance down the ramp, and used a water clock to measure the time taken to move the known distance; this clock was
The book also contains a discussion of infinity. Galileo considers the example of numbers, and their squares. He starts by noting that it cannot be denied that there are as many as there are numbers because every number is a root of some square:
1 <--> 1, 2 <--> 4, 3 <--> 9, 4 <--> 16, and so on.
(In modern terms, it is possible to have a one-to-one correspondence between the elements of a set N and the elements of a proper subset S of N). But he notes what appears to be a contradiction: Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers.
He resolves the contradiction by denying the possibility of comparing infinite numbers: we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," greater," and "less," are not applicable to infinite, but only to finite, quantities. Indeed, he denies that an infinite quantity can meaningfully be said to be greater than a finite quantity. This is a possible resolution, and it implicitly recognises that he has no definition of comparison for infinite numbers, but is less powerful than the modern resolution.
How does this arise in the dialogues? Galileo is discussing problems arising from rolling circles: if two concentric circles of different radius roll along lines, then if the larger does not slip, it appears clear that the smaller must slip. But in what way? Galileo attempts to clarify the matter by considering hexagons, and then extending to rolling polygons with 100,000 sides, or any arbitrarily large number, where he shows that a finite number of finite slips occur on the inner shape. Eventually he concludes that The line traversed by the larger circle consists then of an infinite number of points which completely fill it; while that which is traced by the smaller circle consists of an infinite number of points which leave empty spaces and only partly fill the line, which would not be considered satisfactory now.