To Explain the World: The Discovery of Modern Science (28 page)

As already mentioned in
Chapter 10
, there was at the time a widely held alternative to the theory that bodies fall with uniform acceleration. According to this other view, the speed that freely falling bodies acquire in any interval of time is proportional to the
distance
fallen in that interval, not to the time.
*
Galileo gives
various arguments against this view,
*
but the verdict regarding these different theories of the acceleration of falling bodies had to come from experiment.

With the distance fallen from rest equal (according to the mean speed theorem) to half the velocity attained times the elapsed time, and with that velocity itself proportional to the time elapsed, the distance traveled in free fall should be proportional to the
square
of the time. (See
Technical Note 25
.) This is what Galileo sets out to verify.

Freely falling bodies move too rapidly for Galileo to have been able to check this conclusion by following how far a falling body falls in any given time, so he had the idea of slowing the fall by studying balls rolling down an inclined plane. For this to be relevant, he had to show how the motion of a ball rolling down an inclined plane is related to a body in free fall. He did this by noting that the speed a ball reaches after rolling down an inclined plane depends only on the
vertical
distance through which the ball has rolled, not on the angle with which the plane is tilted.
*
A freely falling ball can be regarded as one that rolls down a vertical plane, and so if the speed of a ball rolling down an inclined plane is proportional to the time elapsed, then the same ought to be true for a freely falling ball. For a plane inclined at a small angle the speed is of course much less than the speed of a body falling freely (that is the point of using an inclined plane) but the two speeds are proportional, and so the distance traveled along
the plane is proportional to the distance that a freely falling body would have traveled in the same time.

In
Two New Sciences
Galileo reports that the distance rolled
is
proportional to the square of the time. Galileo had done these experiments at Padua in 1603 with a plane at a less than 2° angle to the horizontal, ruled with lines marking intervals of about 1 millimeter.
³
He judged the time by the equality of the intervals between sounds made as the ball reached marks along its path, whose distances from the starting point are in the ratios 1
²
= 1 : 2
²
= 4 : 3
²
= 9, and so on. In the experiments reported in
Two New Sciences
he instead measured relative intervals of time with a water clock. A modern reconstruction of this experiment shows that Galileo could very well have achieved the accuracy he claimed.
⁴

Galileo had already considered the acceleration of falling bodies in the work discussed in
Chapter 11
, the
Dialogue Concerning the Two Chief World Systems.
On the Second Day of this previous
Dialogue
, Salviati in effect claims that the distance fallen is proportional to the square of the time, but gives only a muddled explanation. He also mentions that a cannonball dropped from a height of 100
braccia
will reach the ground in 5 seconds. It is pretty clear that Galileo did not actually measure this time,
⁵
but is here presenting only an illustrative example. If one
braccio
is taken as 21.5 inches, then using the modern value of the acceleration due to gravity, the time for a heavy body to drop 100
braccia
is 3.3 seconds, not 5 seconds. But Galileo apparently never attempted a serious measurement of the acceleration due to gravity.

The “Fourth Day” of
Dialogues Concerning Two New Sciences
takes up the trajectory of projectiles. Galileo’s ideas were largely based on an experiment he did in 1608
⁶
(discussed in detail in Technical Note 26). A ball is allowed to roll down an inclined plane from various initial heights, then rolls along the horizontal tabletop on which the inclined plane sits, and finally shoots off into the air from the table edge. By measuring the distance traveled when the ball reaches the floor, and by observation
of the ball’s path in the air, Galileo concluded that the trajectory is a parabola. Galileo does not describe this experiment in
Two New Sciences
, but instead gives the theoretical argument for a parabola. The crucial point, which turned out to be essential in Newton’s mechanics, is that each component of a projectile’s motion is separately subject to the corresponding component of the force acting on the projectile. Once a projectile rolls off a table edge or is shot out of a cannon, there is nothing but air resistance to change its horizontal motion, so the horizontal distance traveled is very nearly proportional to the time elapsed. On the other hand, during the same time, like any freely falling body, the projectile is accelerated downward, so that the vertical distance fallen is proportional to the square of the time elapsed. It follows that the vertical distance fallen is proportional to the square of the horizontal distance traveled. What sort of curve has this property? Galileo shows that the path of the projectile is a parabola, using Apollonius’ definition of a parabola as the intersection of a cone with a plane parallel to the cone’s surface. (See
Technical Note 26
.)

The experiments described in
Two New Sciences
made a historic break with the past. Instead of limiting himself to the study of free fall, which Aristotle had regarded as natural motion, Galileo turned to artificial motions, of balls constrained to roll down an inclined plane or projectiles thrown forward. In this sense, Galileo’s inclined plane is a distant ancestor of today’s particle accelerators, with which we artificially create particles found nowhere in nature.

Galileo’s work on motion was carried forward by Christiaan Huygens, perhaps the most impressive figure in the brilliant generation between Galileo and Newton. Huygens was born in 1629 into a family of high civil servants who had worked in the administration of the Dutch republic under the House of Orange. From 1645 to 1647 he studied both law and mathematics at the University of Leiden, but he then turned full-time to mathematics and eventually to natural science. Like Descartes, Pascal, and Boyle, Huygens was a polymath, working on a wide range of problems
in mathematics, astronomy, statics, hydrostatics, dynamics, and optics.

Huygens’ most important work in astronomy was his telescopic study of the planet Saturn. In 1655 he discovered its largest moon, Titan, revealing thereby that not only the Earth and Jupiter have satellites. He also explained that Saturn’s peculiar noncircular appearance, noticed by Galileo, is due to rings surrounding the planet.

In 1656–1657 Huygens invented the pendulum clock. It was based on Galileo’s observation that the time a pendulum takes for each swing is independent of the swing’s amplitude. Huygens recognized that this is true only in the limit of very small swings, and found ingenious ways to preserve the amplitude-independence of the times even for swings of appreciable amplitudes. While previous crude mechanical clocks would gain or lose about 5 minutes a day, Huygens’ pendulum clocks generally gained or lost no more than 10 seconds a day, and one of them lost only about ½ second per day.
⁷

From the period of a pendulum clock of a given length, Huygens the next year was able to infer the value of the acceleration of freely falling bodies near the Earth’s surface. In the
Horologium oscillatorium
—published later, in 1673—Huygens was able to show that “the time of one small oscillation is related to the time of perpendicular fall from half the height of the pendulum as the circumference of a circle is related to its diameter.”
⁸
That is, the time for a pendulum to swing through a small angle from one side to the other equals π times the time for a body to fall a distance equal to half the length of the pendulum. (Not an easy result to obtain as Huygens did, without calculus.) Using this principle, and measuring the periods of pendulums of various lengths, Huygens was able to calculate the acceleration due to gravity, something that Galileo could not measure accurately with the means he had at hand. As Huygens expressed it, a freely falling body falls 15
¹
/
₁₂
“Paris feet” in the first second. The ratio of the Paris foot to the modern English foot is variously estimated as between 1.06 and 1.08; if we take 1 Paris foot to equal 1.07
English feet, then Huygens’ result was that a freely falling body falls 16.1 feet in the first second, which implies an acceleration of 32.2 feet/second per second, in excellent agreement with the standard modern value of 32.17 feet/second per second. (As a good experimentalist, Huygens checked that the acceleration of falling bodies actually does agree within experimental error with the acceleration he inferred from his observations of pendulums.) As we will see, this measurement, later repeated by Newton, was essential in relating the force of gravity on Earth to the force that keeps the Moon in its orbit.

The acceleration due to gravity could have been inferred from earlier measurements by Riccioli of the time for weights to fall various distances.
⁹
To measure time accurately, Riccioli used a pendulum that had been carefully calibrated by counting its strokes in a solar or sidereal day. To his surprise, his measurements confirmed Galileo’s conclusion that the distance fallen is proportional to the square of the time. From these measurements, published in 1651, it could have been calculated (though Riccioli did not do so) that the acceleration due to gravity is 30 Roman feet/second per second. It is fortunate that Riccioli recorded the height of the Asinelli tower in Bologna, from which many of the weights were dropped, as 312 Roman feet. The tower still stands, and its height is known to be 323 modern English feet, so Riccioli’s Roman foot must have been 323/312 = 1.035 English feet, and 30 Roman feet/second per second therefore corresponds to 31 English feet/second per second, in fair agreement with the modern value. Indeed, if Riccioli had known Huygens’ relation between the period of a pendulum and the time required for a body to fall half its length, he could have used his calibration of pendulums to calculate the acceleration due to gravity, without having to drop anything off towers in Bologna.

In 1664 Huygens was elected to the new Académie Royale des Sciences, with an accompanying stipend, and he moved to Paris for the next two decades. His great work on optics, the
Treatise on Light
, was written in Paris in 1678 and set out the wave
theory of light. It was not published until 1690, perhaps because Huygens had hoped to translate it from French to Latin but had never found the time before his death in 1695. We will come back to Huygens’ wave theory in
Chapter 14
.

In a 1669 article in the
Journal des Sçavans
, Huygens gave the correct statement of the rules governing collisions of hard bodies (which Descartes had gotten wrong): it is the conservation of what are now called momentum and kinetic energy.
¹⁰
Huygens claimed that he had confirmed these results experimentally, presumably by studying the impact of colliding pendulum bobs, for which initial and final velocities could be precisely calculated. And as we shall see in
Chapter 14
, Huygens in the
Horologium oscillatorium
calculated the acceleration associated with motion on a curved path, a result of great importance to Newton’s work.

The example of Huygens shows how far science had come from the imitation of mathematics—from the reliance on deduction and the aim of certainty characteristic of mathematics. In the preface to the
Treatise on Light
Huygens explains:

There will be seen [in this book] demonstrations of those kinds which do not produce as great a certitude as those of Geometry, and which even differ much therefrom, since whereas the Geometers prove their Propositions by fixed and incontestable Principles, here the Principles are verified by the conclusions to be drawn from them; the nature of these things not allowing of this being done otherwise.
¹¹

It is about as good a description of the methods of modern physical science as one can find.

In the work of Galileo and Huygens on motion, experiment was used to refute the physics of Aristotle. The same can be said of the contemporaneous study of air pressure. The impossibility of a vacuum was one of the doctrines of Aristotle that came into question in the seventeenth century. It was eventually understood that phenomena such as suction, which seemed to arise from
nature’s abhorrence of a vacuum, actually represent effects of the pressure of the air. Three figures played a key role in this discovery, in Italy, France, and England.

Well diggers in Florence had known for some time that suction pumps cannot lift water to a height more than about 18
braccia
, or 32 feet. (The actual value at sea level is closer to 33.5 feet.) Galileo and others had thought that this showed a limitation on nature’s abhorrence of a vacuum. A different interpretation was offered by Evangelista Torricelli, a Florentine who worked on geometry, projectile motion, fluid mechanics, optics, and an early version of calculus. Torricelli argued that this limitation on suction pumps arises because the weight of the air pressing down on the water in the well could support only a column of water no more than 18
braccia
high. This weight is diffused through the air, so any surface whether horizontal or not is subjected by the air to a force proportional to its area; the force per area, or
pressure
, exerted by air at rest is equal to the weight of a vertical column of air, going up to the top of the atmosphere, divided by the cross-sectional area of the column. This pressure acts on the surface of water in a well, and adds to the pressure of the water, so that when air pressure at the top of a vertical pipe immersed in the water is reduced by a pump, water rises in the pipe, but by only an amount limited by the finite pressure of the air.