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The Franke Program in Science and the Humanities. Whitney Humanities Center, Yale University. Galileo, Mathematics, and the Arts. January 18, 2012. Mark A. Peterson. Archimedes and the crown: Vitruvius ’ version . Note spillage!.
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The Franke Program in Science and the Humanities Whitney Humanities Center, Yale University Galileo, Mathematics, and the Arts January 18, 2012 Mark A. Peterson
Archimedes and the crown: Vitruvius’ version Note spillage! NBC magazine advertisement, 1940s, Print and Picture Department, Free Library of Philadelphia
Archimedes and the crown: Galileo’s version (ca. 1586) From Galileo and the Scientific Revolution, Laura Fermi and Gilberto Bernardini(1961).
The Pythagorean theory of music, according to traditional sources TheoricaMusicae (1492), FranchinoGaffurio
Pythagorean experiment (ca. 1588) Galileo and his father both report the result of an experiment on the relationship between tension (weight) on a string and the pitch of the plucked string. Their result, unknown before them, is that doubling the weight does not raise the pitch an octave, the way halving the length would do. One must quadruple the weight, and in general the frequency goes up like the square of the tension, and not simply like the tension.
Botticelli: Inferno (ca. 1490) From H. Bredekamp, GalileiderKuenstler(2009).
Brunelleschi’s Dome: a scale model of Dante’s Inferno? (ca. 1588)
The Geometric Compass as Sighting Device (1606) Although Galileo was by now formally an astronomer, writing a comprehensive description of all the things his compass could do, he never discusses sighting on a celestial object. Galileo Opere II, p. 417.
How does strength of a structure behave under scaling? (TNS, 1638)
Two New Sciences: more days Among Galileo’s last (unpublished) works is a dialogue exploring proportionality, Euclid’s Book V, Definition 5, intended by him as a sequel to TNS, a 5th day. A 6th day uses the balance (a sure way to establish proportionality) as an instrument to measure the force (percossa) of falling water.
Euclid’s Book V, Definition 5 Magnitudes are said to be in the same proportion, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Balance when mA=nBfor two integers m and n To prove: mb=na i.e., the weights and the lengths are in the same proportion (at least in the rational case).
and now notice where the center of the system is: Thus mb=na, QED! (And further, not proved here, when b is too big, i.e., mb>na, then B goes down.)
Balance when A’ and B are incommensurate Take A’:B=b:a’ and suppose that, contrary to the rational case, the lever does not balance, but rather that A’ goes down.
[Remember: A’:B=b:a’] • If A’ goes down, remove a small amount E from A’ to make (A’-E):B rational. That is, • m(A’-E)=nB for some integers m and n.
[Remember: A’:B=b:a’] • If A’ goes down, remove a small amount E from A’ to make (A’-E):B rational. That is, • m(A’-E)=nB for some integers m and n. • Then mA’>nB, so mb>na’ (by Definition 5).
[Remember: A’:B=b:a’] • If A’ goes down, remove a small amount E from A’ to make (A’-E):B rational. That is, • m(A’-E)=nB for some integers m and n. • Then mA’>nB, so mb>na’ (by Definition 5). • Thus b is too big, and B goes down, i.e. A’-E goes up. But A’ by itself went down, a contradiction.
It is absurd that an arbitrarily small removal E should cause this change. Thus it cannot be that A’ goes down, and by a similar argument, it cannot be that B goes down. Thus if • A’:B=b:a’, • then A’ and B are in balance. QED.
