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11Math 1700 – Galileo

GalileoAnti-anti-Copernican,

Astronomer, Mathematical Physicist, Founder of Experimental Science


Galileo Galilei1564-1642

Born in Pisa, same year as Shakespeare.Son of a prominent musician and music theorist.Enrolled at University of Pisa to study medicine, 1581

Hated it.


Galileo, the young mathematics whiz

While supposedly studying medicine, Galileo became interested in mathematics and began to study it with a tutor

Left university without a degree in 1585

Appointed professor of mathematics at Pisa in 1589 at age of 25

Salary 1/10 of that of a professor of philosophyGalileo loathed philosophers

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Leaning Tower ExperimentGalileo gathered a group of academics and townspeople at the Leaning Tower of Pisa to witness a demonstration.He dropped metal balls of different weights simultaneously from the tower to demonstrate that Aristotle’s assertion that they landed at different times was wrong


Mathematician in the 16th centuryThe profession of mathematician was just evolving in the 16th century and had two different senses:

1. Calculating magician – like Kepler, whose job included casting horoscopes and uncovering the secrets of nature

2. Precision engineer – someone who knew how to aim the cannon, and could make precision instruments. In short, an engineer. This is what Galileo was.


Galileo proves his worthGalileo set up a workshop at the University of Pisa where he invented, made, and sold instruments for industrial and military purposes.

With this he supplemented his meagreincome and developed a reputation as a fine craftsman and accurate mathematician.

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Galileo gets better jobGalileo parlayed his reputation at Pisa into a better job.He was appointed professor of mathematics at the University of Padua in 1592 for 3 times his Pisa salary.

Padua was the science university of the day.Light teaching duties: Euclid and Ptolemy

Galileo won over to Copernican theory because it could explain the tides.


The TelescopeSpyglass invented in Holland in 1609, magnification of 3 timesGalileo sets out to make his own telescope

First makes an instrument with 8x magnificationThen 20xUltimately 30x

Sold telescopes to Venice merchants to spot ships at sea coming into harbour


Galileo, the AstronomerEventually, Galileo turned the telescope on the heavens to see if they would look different.To his amazement, they did, and he saw many things he could not see before.

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The Starry MessengerIn 1610, Galileo wrote and published a short pamphlet called Siderius Nuncius (The Starry Messenger) in which he reported his amazing findings.


Mountains on the MoonGalileo found that the Moon was not a smooth, perfect sphere, but had an uneven surface.

Mountains as large as the largest on Earth (that Galileo knew of).Determined by measuring shadows cast when the Sun shone at different angles.


EarthshineGalileo found that with a telescope he could make out some features on the dark side of the moon, which was lit by light reflected from the Earth.

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The Medicean “Stars”After repeated nights of observing Jupiter, Galileo noticed that there were four “stars” that appeared to circle round it.

These he called, in honour of the Duke of Tuscany, “Medicean Stars.”They were satellites of Jupiter, showing that the Earth was not the only planet with a “Moon.”

East West

Jan. 7, 1610 * * O *Jan. 8 O * * *Jan. 10 * * O Jan. 11 * * O Jan. 12 * * O *Jan. 13 * O * * *Feb. 26 early * O *Feb. 26 later * O *Feb. 27 * * O * *Feb. 28 * O *Feb. 28 later * * O *March 1 * * * * O


Many More StarsGalileo saw that the “Milky Way” was not just a smear in the sky, but huge clusters of stars.The familiar constellations were surrounded by many stars not visible without the telescope. (Right: Orion’s belt and sword as seen by Galileo.)


The Phases of Venus

Copernican theory implied that Venus should show phases, like the Moon. They were not visible with the naked eye.Galileo could see them with the telescope.

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Sun spotsIn a later work, Galileo reported that he could see spots on the Sun.Also, the Sun rotated, just like the Earth (in the Copernican theory).Galileo had become a committed Copernican.

Galileo’s drawing of sunspots


Evidence for Copernicus’ theory from the telescope

Instead of the heavenly bodies being perfect, smooth, and spherical and able to reflect light, while the Earth is rough and uneven, Galileo showed that the moon has a rough surface and the Earth reflects light onto the Moon. Even the Sun has blemishes and turns, like the Earth.

The Earth is not the only planet with a satellite.

There is much more to the heavens than can be seen without a telescope, suggesting the heavens are vast (as Copernicus stated).

The phases of Venus provide visible confirmation for Copernicus.


Galileo takes another jobThe fame that came to Galileo from The Starry Messenger enabled him to move from the University of Padua to a full-time research position.Galileo accepted an offer to become the Imperial Mathematician to the Duke of Tuscany in Florence in 1610.

No teaching, just research.

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Galileo, the Anti-anti-CopernicanGalileo, armed with evidence from the telescope, took on the Scholastic Philosophers, who defended Aristotle and Ptolemy dogmatically.Galileo, a devout Catholic, began to undermine Aristotle at the privileged interpreter of the Bible.


Letter to the Grand Duchess Christina1615

The Bible uses figurative languageJoshua commanding the Sun to stand still—merely a convenient expression.Anyway, literally, Joshua should have commanded the celestial sphere to stop turning.

Galileo objected to taking quotations out of the Bible, out of context, and believing them to be statements about the natural world.


Galileo instructed not to “hold or defend” the Copernican view

This was the Counter-Reformation.The Catholic Church was most concerned about the Protestant threat to Papal authority.No allowance for different points of view.

In 1616, Galileo was enjoined not to hold or defend the view that the Earth moves and is not in the centre of the world.

Galileo interpreted this as meaning that he was not to say that Copernicus is correct.

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Dialogue Concerning the Two Chief World Systems

1632Platonic style dialogueExplanation of the Tides, from Copernican and Ptolemaic viewpoints--Three characters:

Salviati = the CopernicanSagredo = impartialSimplicio = Aristotelian


The Dialogue, 2Dialogue systematically refutes every tenet of Aristotelian cosmology.

Simplicio’s trump card played on the last page: God can do what He wants.

Galileo called before Inquisition.1633, condemned to life imprisonment for vehement suspicion of heresy.


Galileo, the experimental physicistBefore Galileo got distracted with the telescope, he had been busy trying to understand the phenomenon of motion, particularly motion of an object in free fall.His demonstration of falling bodies at the Leaning Tower of Pisa showed that Aristotle’s theory of how objects fell was incorrect. Galileo wanted to know what was correct.He returned to this in earnest after he was placed under house arrest.

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Projectile MotionMotion of all sorts was a scientific topic of great interest in the Renaissance.A particular problem was the explanation of projectile motion – objects flying through the air.Ancient science tried to explain all motion as either

Some sort of stability, e.g. focus on planetary orbits rather than on the moving planet itself.Or, as the result of direct contact with a motive force.


Aristotle on motionAristotle divided motions on Earth into three categories:

Natural motion – objects seeking their natural place.Forced motion – objects being pushed or pulled.Voluntary motion – objects moving themselves, e.g. animals.

These reasonable categories ran into difficulty explaining projectile motion.


Aristotle’s Antiperistasis

A projectile, e.g., an arrow, was not deemed capable of voluntary motion. Therefore its motion must be either natural or forced (or a combination). Natural motion would take the (heavy) arrow down to the ground. Forced motion required direct contact.Solution: Antiperistasis. The flying arrow divided the air before it, which rushed around to the back of the arrow and pushed it forward.

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The Search for the Aristotelian Explanation

Aristotle’s explanation was so unsatisfactory that Scholastic philosophers through the Middle Ages tried to find a bettter explanation.Impetus theory.

The idea that pushing (or throwing, shooting, etc.) an object imparted something to it that kept it moving along.But what? How? Material? Non-material?


Niccolo Tartaglia, again1500-1550

The man who solved the cubic equation, also concerned himself with projectiles, in particular,cannon balls.Wrote The New Science

Analyzed the path of cannon ballsFound that a cannon will shoot farthest aimed at 45 degrees

Was the teacher of Galileo's mathematics teacher


The Goal of Science: How, not WhyAristotelian philosophy had as its goal to explain everything.

To Aristotle, a causal explanation was not worth much unless it could explain the purpose served by any thing or action.

E.g. A heavy object fell in order to reach its natural place, close to the centre of the universe.

Galileo argued for a different goal for science:Investigate How phenomena occur; ignore Why.Note: This gives paramount importance to the mathematical formulation of phenomena, a major shift for science.

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Galileo on Falling BodiesThe Leaning Tower demonstration showed that Aristotle was wrong in principle about heavier bodies falling faster than lighter ones.

Actually, they did, but only slightly.

Galileo applied Archimedes' hydrostatic principle to motion.

Denser objects fall faster because less buoyed by air.Hypothesis: In a vacuum a feather would fall as fast as a stone.

How could this be tested?


The Idealized ExperimentProblem of testing nature:

Getting accurate measurements.Nature's imperfections interfere with study of natural principles.

Solution: Remove imperfections to the extent possibleMake a nearly perfect model on a human scale (to aid measurement).


Galileo's inclined plane: The first scientific laboratory instrument

To study falling bodies, Galileo invented a device that would slow the fall enough to measure it.Polished, straight, smooth plane with groove, inclined to slow the downward motion as desired.Smooth, round ball, as perfectly spherical as possible.

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Galileo's inclined plane experiment

Roll a smooth, round ball down a polished, straight, smooth path. Incline the path as desired to slow or speed up the fall of the ball.Results: A fixed relationship between distance rolled and time required.


The amazing resultsWhat astounded Galileo was that he found a simple numerical relationship between the distance the ball rolled down the plane and the time elapsed.

n2 x d(nth odd number) x d

n

25d9d5th

16d7d4th

9d5d3rd

4d3d2nd

1d1d1st

Total Distance

Distance rolled in interval n

Time interval


The amazing resultsNo matter how steep or not the inclined plane was set and no matter whether the ball rolled was heavy or light, large or small, it gained speed at the same uniform rate.Also the total distance travelled was always equal to the distance travelled in the first time interval times the square of the number of time intervals.

n2 x d(nth odd number) x d

nth

25d9d5th

16d7d4th

9d5d3rd

4d3d2nd

1d1d1st

Total Distance

Distance rolled in interval n

Time interval

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Galileo’s Law of Uniform Acceleration of Falling Bodies

By concentrating on measuring actual distances and time, Galileo discovered a simple relationship that accounted for bodies “falling” toward the Earth by rolling down a plane.Since the relationship did not change as the plane got steeper, Galileo reasoned that it held for bodies in free fall.


Galileo’s Law of Uniform Acceleration of Falling Bodies, 2

The law states that falling bodies gain speed at a constant rate, and provides a formula for calculating distance fallen over time once the starting conditions are known.Nowhere does the law attempt to explain why a heavy body falls down.The law specifies how a body falls, not why.


Examples:1. A ball is rolled down a plane and travels 10 cm in the first second. How far does it travel in the third second?

Answer: It travels 5 x 10 cm = 50 cm in the third second.5 is the third odd number. 10 cm is the original distance in the first unit of time, which happens to be one second in this case.

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Examples:2. A stone is dropped off a cliff. It falls 19.6 meters in the first 2 seconds. How far does it fall altogether in 6 seconds?

19.6 meters is the unit of distance. The unit of time is 2 seconds. Six seconds represents 3 units of time.The total distance fallen is 32x19.6 meters = 9x19.6 meters = 176.4 meters.


Examples3. Aristotle knew that bodies fall faster and faster over time, but how much faster he could not determine.If an object falls 16 feet in the first second after it is released, how much speed does it pick up as it falls?

Answer: Every second, the object adds an additional 2 x the original distance travelled in the first second to that travelled in the second before it (1d, 3d, 5d, etc.) So in this example, the object accelerates at 2x16=32 feet per second.


What about projectiles?Galileo had devised an apparatus to study falling bodies, based on the assumption that whatever it was that made bodies fall freely through the air also made them roll downhill.How could he make comparable measurements of a body flying through the air?

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Solution: Use the inclined plane again

Since Galileo could measure the speed that a ball was moving when it reached the bottom of his inclined plane, he could use the plan to shoot a ball off a table at a precise velocity.Then he could measure where it hit the floor when shot at different speeds.


Galileo’s trials and calculationsGalileo’s surviving notebooks show that he performed experiments like these again and again looking for the mathematical relationship he thought must be there.


Galileo’s Law of Projectile MotionFinally he found the key relationship:A projectile flying through the air has two distinct motions:

One is its falling motion, which is the same as if it had been dropped. (Constantly accellerating.)The other is the motion given to it by whatever shot it into the air. This remains constant until it hits the ground.

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Another simple solutionThe falling speeds up constantly, the horizontal speed remains the same.Shoot a bullet horizontally at a height of 4.9 meters from the ground and at the same time, drop a bullet from the same height.

They both hit the ground at the same time –one second later.


Many questions answered hereGalileo’s fellow mathematician/engineers were losing a lot of sleep trying to figure out how a cannon fires, how to aim it, etc.Galileo’s Law of Projectile Motion provides a way to solve their problems.


ExampleFrom the top of a cliff, 78.4 meters high, a cannon is shot point blank (horizontally) off the cliff. In the first second it drops 4.9 meters vertically and travells 100 meters horizontally. How far from the base of the cliff will it land?

First figure when it will land. How long will it take to fall 78.4 meters? 78.4 meters = 42 x 4.9 meters. This indicates that it will take 4 seconds to hit the ground.In 4 seconds, the bulled will travel 4 x 100 meters horizontally. It will therefore hit the ground 400 meters from the base of the cliff.

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Galileo’s Two New SciencesGalileo’s work on the science of motion was published in 1638, while under house arrest, and blindThe title was Discourses and Mathematical Demonstrations on Two New Sciences.

One science was motion of bodies (free fall and projectile).The other was strength of materials (an engineering topic).

The book became a model treatise for how to do science. It is the first important work in physics as we know it today.


Galileo's Scientific Method

1. Examine phenomena.2. Formulate hypothesis about underlying

structure.3. Demonstrate effects geometrically.

I.e., give a mathematical account of the phenomena, or “save the phenomena”

4. Calculate the effects expected.5. [Implied.] Compare calculated effects

with observed effects.


Mathematics: The Language of NatureGalileo’s use of mathematics in scientific investigation is different from his predecessors and contemporaries.For Pythagoras, and by analogy, for Plato, Copernicus, and Kepler, mathematics is the secret of nature. To discover the mathematical law is to know what there is to know.For Galileo, mathematics is merely a tool, but an essential one. Nature, he believed, operated in simple relationships that could be described in concise mathematical terms. Mathematics is the Language of Nature.

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