Galileo: Two New Sciences

Galileo Galilei lived in Italy about 400 years ago. He loved to ask simple questions and then test them. One question was very old: if you drop a heavy ball and a light ball at the same moment, which one lands first?

For almost 2,000 years, many teachers followed the Greek thinker Aristotle, who wrote that heavier things fall faster. Galileo thought about it carefully. He said: tie a big stone to a small stone. If small things are slow, the small stone should hold the big one back. But the two stones tied together are heavier than the big stone alone, so they should fall even faster. Both answers can't be right! So the idea must be wrong. Heavy and light things fall together, as long as the air doesn't get in the way.

Falling is fast, too fast to time with the clocks of his day. So Galileo slowed it down. He let a bronze ball roll down a long, smooth, gently sloping board, and timed it with water dripping into a cup. He found a pattern: in 2 beats of time the ball went 4 times as far as in 1 beat. In 3 beats it went 9 times as far.

You may have heard that Galileo dropped balls from the Leaning Tower of Pisa. That story was told years after he died, and no one knows if it happened. What we do know is that he did many careful rolling-ball tests, and he wrote them down.

Who, when and why

Galileo Galilei (1564–1642) taught mathematics at Pisa and Padua and later served the Grand Duke of Tuscany. He is famous for turning a telescope on the sky in 1609 and 1610. In 1633 the Roman Inquisition tried him over his Dialogue on the Two Chief World Systems, which favored the sun-centered view; it found him "vehemently suspected of heresy" and confined him to his house near Florence. There, old and losing his sight, he finished the book he thought was his most important: Discourses and Mathematical Demonstrations Concerning Two New Sciences. It was printed in 1638 by the Elzevir firm in Leiden, in the Dutch Republic.

The book is a conversation over four days among three friends. Salviati speaks for Galileo. Sagredo is a quick, curious gentleman. Simplicio defends the teaching of Aristotle. The two "new sciences" are the strength of materials (why big things break more easily than small ones built the same way) and motion (falling bodies and thrown ones).

The problem Galileo faced

Aristotle taught that a heavy body falls faster than a light one, in proportion to its weight, and that every motion needs something pushing it. Scholars had argued about this for centuries. Some, like the 14th-century thinkers at Oxford and Paris, had already worked out the math of steadily speeding-up motion. Galileo's problem was to show what falling bodies actually do and to describe it in numbers, when falls are over in a second or two and there were no stopwatches.

Idea 1: heavy and light fall together

Galileo gives two kinds of reason. The first is an argument you can do in your head, a thought experiment. Suppose heavier means faster. Tie a fast big stone to a slow small one. The slow one should drag the fast one back, so the pair goes slower than the big stone. But the pair is heavier than the big stone, so it should go faster. One assumption has led to two opposite answers, so the assumption is false.

The second is the role of the medium, the air or water a body falls through. Galileo argued that differences in falling come from the medium's resistance, not from weight itself. In a vacuum, he reasoned, all bodies would fall at the same speed. He could not make a good vacuum; Robert Boyle's air pump later showed a coin and a feather falling together, and in 1971 the astronaut David Scott dropped a hammer and a feather on the Moon, where they landed together.

Idea 2: the law of fall

On the Third Day Galileo defines uniformly accelerated motion: equal amounts of speed are added in equal times. From that definition he proves that the distance covered from rest grows as the square of the time. Double the time, four times the distance. In equal time steps the distances go as the odd numbers 1, 3, 5, 7, …

To test it he slowed the motion down with a ramp. A ball rolling down a slope speeds up the same way a falling body does, only more gently.

For a clock he let water run from a large vessel through a thin pipe into a glass during each roll, then weighed the water on a balance. The weight of water stood for the time.

Idea 3: thrown things follow a parabola

On the Fourth Day Galileo puts two motions together. A ball rolled off a table keeps its sideways speed (nothing slows it, if we ignore air), while it falls with the law of squares. Steady sideways motion plus speeding-up downward motion traces a parabola. He used this to make tables for gunners: the range is greatest at an aim of 45°, and aims equally above and below 45° (say 30° and 60°) give the same range.

Idea 4: why giants can't be built like us

The First and Second Days are about the strength of beams, ropes and bones. Galileo noticed a law of size. If you make a beam twice as long, twice as wide and twice as thick, its weight grows 2 × 2 × 2 = 8 times, but the area of its cross section, which carries the load, grows only 2 × 2 = 4 times. So big structures are weaker for their size. That is why, he said, a giant twice our height would need much thicker bones, and why a small dog can carry two or three dogs its size on its back while a horse could not carry even one horse.

Getting the history right

• The Leaning Tower. The famous story comes from a life of Galileo written by his last student, Vincenzo Viviani, in the 1650s, more than sixty years after the event was supposed to happen. Galileo never mentions it. Historians treat it as uncertain at best. In 1586 the Flemish engineer Simon Stevin and Jan Cornets de Groot really did drop two lead balls, one ten times heavier, from a church tower in Delft, and heard them land together.
• The ramp experiments were real. Some 20th-century writers doubted that Galileo could get such clean results. In 1961 Thomas Settle rebuilt the grooved board and water clock and got results as good as Galileo claimed, and Stillman Drake found Galileo's own working notes with measured distances (around 1604–1608).
• Not out of nowhere. The "mean speed" rule for steadily speeding-up motion was proved in the 1300s by the Oxford Calculators and drawn as a diagram by Nicole Oresme. Galileo's new step was to say that real falling bodies move this way and to test it.

What later science kept, and what it changed

Kept: the law of fall, the idea that heavy and light bodies fall alike without air, the parabola of a thrown body, the independence of sideways and downward motion, and the square–cube law of size. Changed: Galileo thought an unpushed body on level ground would keep moving, but he pictured "level" as a circle around the Earth. Descartes and then Newton stated the law of inertia for straight lines, and Newton explained why bodies speed up: a force, gravity, proportional to mass, which is why every mass gets the same acceleration. Air resistance, which Galileo set aside, is now worked out with equations he did not have.

Reasoning and experiment together. Galileo is often called the father of experimental science, but his book leans as hard on mathematics and imagined cases as on measurement. He used experiments to check a mathematical definition he had chosen because it was simple. Compare Francis Bacon, who wanted to gather instances first, and Descartes, who wanted to start from first principles. Which way does science actually work? Most scientists today would say: all three, back and forth.

The puzzle of infinity. On the First Day Salviati notices something strange. Every whole number has a square, so there seem to be as many squares (1, 4, 9, 16, …) as numbers. Yet most numbers are not squares. Galileo concluded that "equal", "greater" and "less" don't apply to infinite collections. Two centuries later Georg Cantor took the opposite path and built a theory in which the squares and the counting numbers are the same size, because they can be paired one to one.

The pendulum. Galileo also noticed that a pendulum's swing takes about the same time whether it swings wide or narrow, and that the length must be four times as great to double the time. This became the basis of the pendulum clock, built by Christiaan Huygens in 1656, which made accurate timekeeping possible for the first time.

Big questions. Can a thought experiment prove something about the real world? When a famous authority and a careful test disagree, what should we believe, and how sure should we be? Why might mathematics, made in our heads, describe falling stones so well?

How later books answered. Newton's Principia (1687) opens with laws of motion that build directly on Galileo, and its third book shows that the same gravity that pulls a falling stone holds the Moon in its orbit. Huygens worked out the math of the pendulum and of circular motion.