Dialogues Concerning Two New Sciences

Overview

Two New Sciences is three hundred pages, cast as a conversation over four days between three men in a Venetian shipyard. Salviati speaks for Galileo, Sagredo is the intelligent layman who asks the right questions, and Simplicio defends Aristotle. They are the same three characters from the Dialogue Concerning the Two Chief World Systems, the book that got Galileo tried and condemned in 1633.

The circumstances of its writing are part of its meaning. Galileo was seventy-four, under permanent house arrest at Arcetri, forbidden to publish anything, and had gone blind. The manuscript was smuggled to the Netherlands and printed at Leiden in 1638 by the Elzevirs, out of reach of the Roman censors, with a preface in which the publisher claims Galileo had nothing to do with it.

There is a common assumption that this was a book about astronomy. It is not. Galileo returned to problems he had worked on at Padua decades earlier and had abandoned when the telescope made him famous, and the subject is terrestrial: why beams break, and how things fall. That turns out to matter far more.

The form is mixed and can be awkward. The first two days are conversation. The third and fourth days are conversation wrapped around a Latin treatise which Salviati reads aloud, with propositions, theorems and proofs in the manner of Euclid, and the others interrupting.

The argument

First Day: cohesion, the vacuum and the infinite. It opens in the arsenal at Venice with a practical puzzle: why do large machines and large structures fail where small ones of the same design do not?

Galileo’s answer, developed on the Second Day, is the square-cube relation. Scale a shape up and the area of its cross-sections grows as the square of the linear dimension while its weight grows as the cube, so strength cannot keep pace with load. A beam twice as long and twice as thick is relatively weaker. A giant built to human proportions would be crushed by his own frame. Nature therefore cannot make trees or animals beyond a certain size without changing their proportions, which is why the bones of large animals are disproportionately thick.

The rest of the First Day ranges widely: what holds matter together, whether a vacuum can resist separation, why a pump cannot lift water beyond about thirty-four feet, the behaviour of pendulums, and a long and genuinely strange excursion into the paradoxes of the infinite. Salviati shows that the squares can be put in one-to-one correspondence with all the integers even though most numbers are not squares, and concludes that the attributes greater, less and equal simply do not apply to infinite quantities. That is Galileo’s paradox, and it is a real anticipation of Cantor.

Also here is the refutation of Aristotle on falling bodies, done by argument rather than by dropping anything. Suppose a heavy body falls faster than a light one. Tie them together. The lighter should retard the heavier, so the combination falls slower than the heavy body alone. But the combination is heavier than either, so it should fall faster. The premise generates a contradiction and must go. Whether Galileo ever dropped weights from the tower at Pisa is doubtful; this argument needs no tower.

Second Day: the strength of materials. A systematic treatment of beams: how a loaded beam breaks, where it breaks, how strength varies with length, thickness and how it is supported, and the shape a beam should have to be equally strong along its length. This is the founding text of the strength of materials, and although some of Galileo’s numbers are wrong, because he misjudged how stress distributes across a section, the questions and the method are right.

Third Day: motion. The heart of the book. Galileo defines uniform motion, then defines uniformly accelerated motion as motion that gains equal increments of speed in equal times, and notes that this is a definition rather than an observation: the question is whether nature uses it.

From the definition he proves the two central results. The mean speed theorem, that a uniformly accelerated body covers the same distance as a body moving throughout at the average of its initial and final speeds. And the times-squared law, that distances from rest are as the squares of the elapsed times, so a body falls one unit in the first interval, three in the second, five in the third, in the odd numbers.

He could not test this with free fall, which is far too fast to time with the water clocks available. So he rolled bronze balls down a grooved inclined plane, slowing the process without changing its character, and argued that what holds on the incline holds in the limit for vertical fall. The passage describing the apparatus, the groove lined with parchment, the water weighed to measure time, and the repetitions, is one of the first detailed accounts of experimental procedure in Western science.

Fourth Day: projectiles. Galileo separates a projectile’s motion into two independent components, a uniform horizontal motion and a uniformly accelerated vertical one, and shows that compounding them yields a parabola. He derives the range, notes that forty-five degrees gives the maximum, and tabulates the results for gunners. The principle that independent motions superpose without interfering is as important as the parabola.

How to read it

Read the First Day for the argument against Aristotle and the square-cube reasoning, the Third Day for the law of fall and the inclined plane, and the Fourth Day for the parabola. The Second Day is for engineers.

Follow the inclined-plane passage closely and ask what is being assumed. Galileo cannot observe free fall, so he measures something else and argues that the law carries over. That inference is doing enormous work and it is worth deciding whether it holds.

Notice the idealisation. Galileo throws away air resistance, friction and the resistance of the medium, on the explicit ground that you have to find the underlying law first and add the complications afterwards. Simplicio objects that real bodies do not behave this way, and he is right, and Galileo is righter.

Read Simplicio charitably. He is set up to lose, but the Aristotelian position he defends is not stupid, and in ordinary experience a feather does fall more slowly than a stone.

Keep the biography in view. A blind old man under house arrest, forbidden to publish, returning to work he had set aside thirty years before and getting it out of the country, is not incidental colour. It is why the book exists.

Questions it raises

• Galileo defines uniform acceleration and then asks whether nature uses that definition. What kind of question is that, and how would you settle it?
• The law of fall is established on inclined planes and asserted for free fall. Is the extrapolation justified?
• He idealises away resistance and friction. When is it legitimate to describe a motion nobody can observe?
• The square-cube argument sets limits on the size of animals and buildings. Why had nobody noticed?
• Galileo’s paradox shows the squares can be matched one-to-one with all the numbers. He concludes that comparison breaks down for infinities. Was there a better conclusion available to him?
• This is a book about beams and falling stones, written by a man condemned over the motion of the earth. Did he choose the subject to be safe, or because he thought it was the more important one?