Galileo, Mathematics, and the Arts

January 18, 2012

Mark A. Peterson

The Franke Program in Science and the Humanities

Whitney Humanities Center, Yale University

NBC magazine advertisement, 1940s, Print and Picture Department, Free Library of Philadelphia

Archimedes and the crown: Vitruvius’ version

Note spillage!

Archimedes and the crown: Galileo’s version (ca. 1586)

From Galileo and the Scientific Revolution, Laura Fermi and Gilberto Bernardini (1961).

Archimedes Law of the LeverA:B=b:a

The Pythagorean theory of music, according to traditional sources

Theorica Musicae (1492), Franchino Gaffurio

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.

Renaissance Lute frets

Botticelli: Inferno (ca. 1490)

From H. Bredekamp, Galilei der Kuenstler (2009).

Manetti’s Inferno (ca. 1490)

Would Manetti’s roof collapse?

Brunelleschi’s Dome: a scale model of Dante’s Inferno? (ca. 1588)

Galileo’s military compass (ca. 1606)

Solving 138:400=x:190

The Geometric Compass as Sighting Device (1606)

Galileo Opere II, p. 417.

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.

Uniform Acceleration in Two New Sciences (1638)

Musical Proportions from TNS (1638)

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Among Galileo’s last (unpublished) works is a dialogue exploring proportionality,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.

Aggiunti’s Oratio (1627)

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.

Archimedes Law of the LeverA:B=b:a

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).

Reshape the masses A and B, keeping their centers fixed …

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.