A Few Good Men A handful of brave men armed with the weapons of mathematics and courage toppled, in a span of a mere one hundred years, the entire geocentric model of the universe. The Polish astronomer Copernicus1 (1473-1543) challenged the geocentric model of Ptolemy (the one with the epicycles) on the grounds that placing the sun at the center of the solar system and assuming that Earth revolves about the sun (and rotates around its axis) reduces the number of equations describing the motion of the planets from about eighty down to thirty. His book De revolutionibus orbium coelestium appeared in 1543 after his death. The Vatican ignored the book as it only suggested that the mathematical model putting the sun at the center makes more sense. He didn’t assert that this is the way things are. At the time of publication of this first round in the cosmic battle, the major hero, Galileo, was not yet born. We shall get to him soon. A Danish astronomer, Tycho Brahe2 (1546-1601), patiently collected a mountain of astronomical data over a ten-year period. Upon his death, his assistant Johan Kepler3 (1571-1630), whom he had taught to observe and then hypothesize, interpreted the data and formulated his three laws of planetary motion. 1Nicole Oresme also opposed the theory of a stationary Earth as proposed by Aristotle and advocated the motion of Earth some 200 years before Copernicus. He eventually rejected his own ideas. 2He was appointed Imperial Mathematician to the Holy Roman Emperor, Rudolph II, and Kepler was hired as his assistant to help with the calculations. He also wore a golden nose to replace his own which he lost in a duel. 3Kepler’s mathematical work on the volume of a wine barrel is considered to be at the forefront of integral calculus and the calculation of volumes of solids of revolution. · 1. The planets revolve about the sun in elliptical orbits, with the sun at one focal point of the ellipse. (An ellipse has two focal points, or foci.) · 2. An imaginary line from the sun to a planet sweeps out equal areas in equal time intervals. · 3. No matter which planet we study, the ratio of the square of the average distance from the sun to the cube of the length of time of one complete revolution is the same. Aha! The motion of the planets is entirely predicable using mathematics. Furthermore, the Church and the ancient philosophers were wrong! The orbits are elliptic – not circular – and Earth is just another planet. And the best part is that these laws rest on mathematics and observation – not on authority. Kepler, too, escaped the wrath of the Roman inquisition. He lived outside of Rome’s sphere of influence. Our main hero, as we shall see, was not as lucky. Galileo4 (1564-1642) was the son of a Florentine merchant. As a boy, he studied music, art, and poetry. He designed mechanical toys. He showed mathematical promise when he was a medical student. He noticed that after the hanging lamps were filled with oil and lit, they swung back.

1. A Few Good Men
A handful of brave men armed with the weapons of mathematics
and courage toppled, in a span of a mere one hundred years, the
entire geocentric model of the universe. The Polish astronomer
Copernicus1 (1473-1543) challenged the geocentric model of
Ptolemy (the one with the epicycles) on the grounds that placing
the sun at the center of the solar system and assuming that Earth
revolves about the sun (and rotates around its axis) reduces the
number of equations describing the motion of the planets from
about eighty down to thirty.
His book De revolutionibus orbium coelestium appeared in 1543
after his death. The Vatican ignored the book as it only
suggested that the mathematical model putting the sun at the
center makes more sense. He didn’t assert that this is the way
things are.
At the time of publication of this first round in the cosmic
battle, the major hero, Galileo, was not yet born. We shall get to
him soon.
A Danish astronomer, Tycho Brahe2 (1546-1601), patiently
collected a mountain of astronomical data over a ten-year
period. Upon his death, his assistant Johan Kepler3 (1571-
1630), whom he had taught to observe and then hypothesize,
interpreted the data and formulated his three laws of planetary
motion.
1Nicole Oresme also opposed the theory of a stationary Earth as
proposed by Aristotle and advocated the motion of Earth some
200 years before Copernicus. He eventually rejected his own
ideas.
2He was appointed Imperial Mathematician to the Holy Roman
Emperor, Rudolph II, and Kepler was hired as his assistant to
help with the calculations. He also wore a golden nose to
replace his own which he lost in a duel.
3Kepler’s mathematical work on the volume of a wine barrel is
considered to be at the forefront of integral calculus and the

2. calculation of volumes of solids of revolution.
· 1. The planets revolve about the sun in elliptical orbits, with
the sun at one focal point of the ellipse. (An ellipse has two
focal points, or foci.)
· 2. An imaginary line from the sun to a planet sweeps out equal
areas in equal time intervals.
· 3. No matter which planet we study, the ratio of the square of
the average distance from the sun to the cube of the length of
time of one complete revolution is the same.
Aha! The motion of the planets is entirely predicable using
mathematics. Furthermore, the Church and the ancient
philosophers were wrong! The orbits are elliptic – not circular –
and Earth is just another planet. And the best part is that these
laws rest on mathematics and observation – not on authority.
Kepler, too, escaped the wrath of the Roman inquisition. He
lived outside of Rome’s sphere of influence. Our main hero, as
we shall see, was not as lucky.
Galileo4 (1564-1642) was the son of a Florentine merchant. As
a boy, he studied music, art, and poetry. He designed
mechanical toys. He showed mathematical promise when he was
a medical student. He noticed that after the hanging lamps were
filled with oil and lit, they swung back and forth in a periodic
way. Although the arcs of these pendulums got progressively
shorter, the amount of time for one full sweep back and forth
remained constant. He filed this away in his young brilliant
mind until, in his old age, he invented the grandfather clock
based on the motion of a pendulum.
4Immortalized forever in Queen’s Bohemian Rhapsody.
In 1589, he obtained a teaching post at the University of Pisa –
a job that he soon lost because he spoke of errors he had found
in Church cosmology. He sought employment in a college not
under the sway of the Vatican.
Before long, he was a teacher in the secular University of Padua
where he continued developing his interest in motion. The

3. university was under the rule of Venice and, therefore,
independent of papal authority.
Galileo was the first thinker to notice that the path of a thrown
object is an upside-down parabola. This was interesting to the
military folks who knew very little about the trajectories of
cannonballs. Galileo showed them that the maximum range
results when the angle of the cannon is 45 °. Furthermore, a
deviation from 45 ° in either direction results in the same
shortening of the range. Thus an angle of 30 ° and an angle of
60 ° will produce the same range, though the trajectories will be
different. (The common deviation here is 15 °.) See Figure 8-
1 to help you picture this.
Figure 8-1
Galileo was a pioneer in the mathematical study of motion in
other ways. He argued that since the distance traveled by an
object moving with constant speed is the product of the speed
and the time(distance = rate × time), one could compute
distance by computing the area of a rectangle in a diagram,
much like the graphs of today, in which a horizontal axis
represents time and a vertical axis represents speed. A carriage
traveling at 20 miles per hour for 10 hours clearly travels 200
miles, which is the area of the rectangle depicted in Figure 8-2,
in which the axes are labeled t and vfor time and velocity,
respectively.
Figure 8-2
Now, said Galileo, suppose the carriage starts with velocity 20
miles per hour (in units of his day) and slows down at a
constant rate to zero miles per hour at the end of the trip. In
technical jargon, the carriage decelerates uniformly. Then we
have the picture in Figure 8-3, and the resulting right triangle
has area × 10 × 20 = 100, meaning that the carriage travels 100
miles.
Figure 8-3

4. Uniform deceleration and acceleration interested Galileo
because his experiments indicated that the velocity of a falling
object accelerates at the rate of 32 feet per second, per second,
that is, each second, the velocity increases by 32 feet per
second. Physicists write this as ft/sec2.
Galileo then proceeded to calculate the distance D that a
dropped body falls in N seconds, as follows. (This argument is
brilliant!) The chart of Figure 8-4 shows the speed of the falling
object after N seconds.
Figure 8-4
Galileo then realized that the average speeds during these
seconds are given by the table of Figure 8-5, where each
average is obtained by adding the speeds at the endpoints of the
time interval and dividing by 2.
Figure 8-5
Then the distance covered in N seconds is just the sum of the
average speeds. Now Galileo saw the pattern of these average
speeds. They are odd multiples of 16. That’s right!
16 = 1 × 16
48 = 3 × 16
80 = 5 × 16
112 = 7 × 16
He then factored out the 16 and obtained, tentatively, that D =
16 × (the sum of the first N odd numbers). Great! But what is
the sum of the first N odd numbers? Fortunately, Galileo was
familiar with the mathematics of ancient Greece. The
Pythagoreans noticed that a square array of dots can be
partitioned as a succession of L-shaped arrays of dots as shown
in Figure 8-6. But each L has anodd number of dots and they are
consecutive! Galileo’s problem was solved two thousand years
before he was born. The sum of the first N odd numbers is N2.
His formula D = 16 × N2 is still used in physics courses today –
with different letters.
Figure 8-6

5. These types of geometric shapes and numbers fascinated the
Pythagoreans. They called themfigurate numbers. Another
example of these, other than the square numbers given above,
are thetriangular numbers 1, 3, 6, 10, …, which come from
arranging dots in triangular arrays seen inFigure 8-7.
Figure 8-7
In 1609, Galileo heard about an invention from far away
Holland that would dramatically affect the course of events in
his life (and in ours) – the telescope. He immediately started to
build one. One night in January 1610, Galileo turned his
recently acquired telescope to the skies and observed several
small white dots on one side of Jupiter (the planet – not the
god). The next night, he saw two dots on that side, and the next
night, saw one dot emerge on the other side of the planet. He
immediately realized that he was seeing moons which were
revolving around Jupiter – and not around the earth – in direct
contradiction to the geocentric theory. He was also aware of
another contradiction to Church (and ancient) cosmology.
Aristotle and the church taught that heavenly bodies were
perfect spheres. But “seeing is believing” – and Galileo saw
craters on our very own moon.
He published his findings, of course. Then to make matters
worse, in 1615, Galileo took his findings to Rome. We guess he
didn’t have good legal advice. He was told to go home and
sternly warned to drop the matter. At this time, the Vatican
decided to suspend Copernicus’ book (then over sixty years
old), realizing the magnitude of the challenge to their
cosmology.
Galileo pressed his luck, as we say, and wrote a book
entitled Dialogues on the Two Principal Systems of the
World in which he ridiculed the Pope and the church-supported
geocentric theory. He was eventually hauled before the Roman
Inquisition, where he was given a chance to recant his heretical
views and save himself from a horrible death at the stake. He

6. recanted on his knees and spent the rest of his life between jail
and house arrest. He wrote another book titled Dialogues
Concerning Two New Sciences. This courageous man’s writings
were smuggled out of the country and published. He was
pardoned by the Vatican posthumously in 1992. (This date is
not a misprint.)
Galileo died in 1642, and the ailing geocentric theory died soon
after. So did the philosophy of the medieval scholastics that
motion must be explained using a teleological approach, that is,
actions towards ends or goals such as “an apple falls to the
ground because it belongs there,” or “things are drawn to the
earth because it is the most important place in the universe,” or
“the rose is red to symbolize the blood of Christ.” Galileo
established the validity of describing nature – mathematically of
course. He substituted “how” for “why.” This set a scientific
trend for physics that is still valid today, although many
theories include causal explanations. Galileo (and Kepler and
others) showed that nature behaves in a mathematically
predictable way, that is, he established the idea of natural law
that we take for granted today but which was hardly obvious
then.
Let us close this chapter with the observation of the English
scientist Francis Bacon,5 who dealt a final deathblow to
medieval scholasticism which confined itself to syllogistic, or
deductive logic, that is, proceeding from the general to the
particular. He pointed out that science must be inductive. It
must formulate general rules after studying particular facts.
Induction works mathematically. An alleged causal factor, A,
must be present when the effect, B, is observed and must be
absent when the effect is not observed. Moreover, A must be
present to a greater degree when B is strong and ought to be
present to a lesser degree when B is weak.
If one claims that it is the weight of a rock dropped on your toe
that causes it to hurt, then a lighter rock should produce a
quieter yelp of pain than a heavier one should. If one claims
that it is the gray color that hurts, the claim is absurd since a

7. gray pebble produces no pain while a heavy brown rock elicits a
(barbaric) yelp.
The fifteenth and sixteenth centuries saw the world turned
upside down – the Reformation, the new world, the printing
press, Earth exiled to the third tier in the solar system. The sails
of change were raised to the wind and the world was ripe for
revolution!