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\begin{document}
\begin{center}
{\bf \large A Tiny Taste of the History of Mechanics \\}
\vspace{0.5cm}
{\small January 8, 2008 \\}
{\small John Baez}
\end{center}
These are some incredibly sketchy notes, designed to convey
just a tiny bit of the magnificent history of mechanics, from
Aristotle to Newton. I've left out huge amounts of important
and interesting stuff.
\subsection*{Aristarchus of Samos}
Aristarchus (310 BC - $\sim$240 BC) argued that the Earth orbited
the Sun. In his book \textsl{On the Sizes and Distances of the
Sun and Moon} he calculated the relative distances of the Sun and
Moon by noting the angle between the Sun and Moon when the Moon was
half full. His logic was correct, but his measurement of the angle
was wrong (87 degrees instead of 89.5), so he concluded that the Sun
was 20 times farther than the Moon, when it's actually about 390
times farther.
Archimedes later wrote:
\begin{quote}
You King Gelon are aware the `universe' is the name given by most
astronomers to the sphere the centre of which is the centre of the
earth, while its radius is equal to the straight line between the
centre of the sun and the centre of the earth. This is the common
account as you have heard from astronomers. But Aristarchus has
brought out a book consisting of certain hypotheses, wherein it
appears, as a consequence of the assumptions made, that the universe
is many times greater than the `universe' just mentioned. His
hypotheses are that the fixed stars and the sun remain unmoved, that
the earth revolves about the sun on the circumference of a circle, the
sun lying in the middle of the orbit, and that the sphere of fixed
stars, situated about the same centre as the sun, is so great that the
circle in which he supposes the earth to revolve bears such a
proportion to the distance of the fixed stars as the centre of the
sphere bears to its surface.
\end{quote}
Archimedes complained that the last sentence was mathematically meaningless.
Recent commentators have suggested that Aristarchus was tyring to say the
sphere of fixed stars was \textit{infinite} in size!
\subsection*{Eratosthenes}
Eratosthenes measured the circumference of the Earth to be
252,000 stadia by comparing shadows at noon in two cities
at different latitudes. Unfortunately we don't know exactly
how long his `stadia' are! So, all we know is that his figure
is within 20\% of the right answer (about 40,000 kilometers).
\subsection*{Aristotle}
Aristotle wrote his \textsl{Physics} as lecture notes sometime
around 350 BC. His work was more philosophical than quantitative
in nature, unlike the previous authors listed. He adopted an
Earth-centered universe.
Some basic principles: All that moves is moved by something
else. Action at a distance is inconceivable: the
mover must always be connected to the moved.
What about falling bodies? This proved a bit embarrassing, since
it seems to violate the above principles.
Aristotle gives no formulas, but at points he seems to suggest
that velocity is proportional to force divided by ``resistance":
$$ v \propto F/R $$
For falling bodies this might mean
$$ v \propto m/R $$
where $m$ is the body's mass.
This is actually \textsl{true} for a body falling at terminal velocity in
air or some other medium with friction, but it doesn't address the
problem of how falling bodies \textsl{accelerate}.
Aristotle believed that the four elements (earth, water, air and fire)
each sought their own proper level, so earth fell towards the center
of the universe, with water above that, then air, then fire. He
believed that the stars and planets were made of a different substance
than terrestrial matter (`aether', later called `quintessence',
meaning `fifth element'). So, it probably never occurred to him to
find a unified theory of motion applying equally to a falling rock and
the moon. This was Newton's huge achievement.
\subsection*{Archimedes}
Archimedes (287 BC - 212 BC) did amazing work on statics --- and as we
now know through recently discovered texts, he essentially invented
the integral calculus.
\subsection*{Ptolemy}
Ptolemy seems to have lived in Alexandria, 83 -- 161 AD. In his
\textsl{Almagest} he described a geocentric solar system with
epicycles to compute the motions of the Sun, Moon, and 5 visible
planets. It's extremely accurate, and makes use of some very
interesting mathematics. Most people who make fun of Ptolemy's
epicycles are complete idiots compared to Ptolemy.
\subsection*{Dark Ages}
When the Romans took over Greece a long process of scientific
decline began, nicely discussed in Lucio Russo's \textsl{The
Forgotten Revolution: How Science Was Born in 300 BC and Why
It Had to Be Reborn}. Most Greek texts were lost completely;
the ones that survived did so through a highly complex process
of repeated translation, discussed in Scott Montgomery's
\textsl{Science in Translation: Movements of Knowledge through
Cultures and Time}. A quote from the review by John Stachel:
\begin{quote}
Perhaps the best of the book's many delightful challenges to
conventional wisdom comes in the first section on the translations of
Greek science. Here we learn why it is ridiculous to use a phrase like
``the Renaissance recovery of the Greek classics''; that in fact the
Renaissance recovered very little from the original Greek and that it
was long before the Renaissance that Aristotle and Ptolemy, to name
the two most important examples, were finally translated into
Latin. What the Renaissance did was to create a myth by eliminating
all the intermediate steps in the transmission. To assume that Greek
was translated into Arabic ``still essentially erases centuries of
history'' (p. 93). What was translated into Arabic was usually Syriac,
and the translators were neither Arabs (as the great Muslim historian
Ibn Khaldun admitted) nor Muslims. The real story involves Sanskrit
compilers of ancient Babylonian astronomy, Nestorian Christian
Syriac-speaking scholars of Greek in the Persian city of Jundishapur,
and Arabic- and Pahlavi-speaking Muslim scholars of Syriac, including
the Nestorian Hunayn Ibn Ishak (809-873) of Baghdad, ``the greatest of
all translators during this era'' (p. 98).
\end{quote}
So, it's important to remember that the `Dark Ages' were dark only
in Western Europe (`Christendom'), and that meantime Muslim scholars
were making progress in astronomy, mathematics and so on. But, when
the West finally got going, it did some wonderful things.
\subsection*{Etienne Tempier, Bishop of Paris}
In the Middle Ages almost all Western European scholars were monks.
The works of Aristotle were introduced around 1200 when they were
imported via Arab sources, for example from Andalusia (now southern
Spain). As they spread through Europe, they caused conflict with
accepted views (a blend of Christian theology and Plato's philosophy).
In 1277 the Bishop of Paris condemned a list of 219 theses, including
some related to physics. Here are some of the condemned theses:
\begin{itemize}
\item
66. That God is unable to to impart rectilinear uniform motion
to the heavens.
\item
102. That nothing happens by chance, but everything comes about
by necessity, and that all the things that will exist in the future
will exist by necessity...
\end{itemize}
Also: if one thing affects another, the second must also affect
the first!
\subsection*{William Occam}
Occam (1288-1347) was a Franciscan friar famous for his `razor';
he also believed that in the absence of resistance motion would
continue indefinitely.
\subsection*{Nicole Oresme}
Oresme (1323-1382) was perhaps the first to draw pictures resembling
graphs, which plot the change of some quantity (or `form') as a
function of time --- though not on a rectilinear grid of 'graph
paper'. This is very important because it's a step towards the
later idea that time is like space.
In his book \textsl{Latitude of Forms} he studied many ways one
quantity could vary as a function of another, including `uniformly
difform' quantities, i.e.\ those that change at a constant rate.
Using these charts he
showed that an object whose speed was `uniformly difform' would move a
distance from time $t_1$ to time $t_2$ equal to $v_{mean}(t_2 - t_1)$
where $v_{mean}$ is the object's speed at a time \textsl{halfway
between $t_1$ and $t_2$}. In modern language: if the acceleration
$$v'(t) = a$$ is constant, the change of position is
$$ \int_{t_1}^{t_2} v(t)\, dt = \frac{1}{2} a (t_2 - t_1)^2 $$
Later people including (but not only) Galileo applied this idea
to falling objects.
Oresme also proved the divergence of the harmonic series!
\subsection*{Nicolaus Copernicus}
In 1543, in his \textsl{De Revolutionibus Orbium Coelestium},
Copernicus rejects Ptolemy's model of the solar system and reverted to
an earlier Greek model in which the Earth goes around the Sun and all
orbits are perfectly circular. This makes predictions much less
accurate than Ptolemy's!
\subsection*{Johannes Kepler}
In 1596, Kepler published his \textsl{Mysterium Cosmographicum},
which adopted a Copernican heliocentric cosmology and attempted
to explain the radii of the planet's orbits in terms of nested
Platonic solids.
Later he spent years analyzing accurate data collected with his boss
Tycho Brahe, and came up with a system where planets moved along
circular orbits \textit{not quite centered at the Sun}. (The
off-center circle idea was already familiar to Ptolemy and called in
Latin a \textit{punctum aequans}.) However, he discovered slight
discrepancies in the orbit of Mars (just 8 minutes, a minute being a
60th of a degree) which eventually led him to discard this system.
In the years that followed, he realized first that the planets could
not move with constant speed around their orbits, and then that the
orbits should be \textit{ellipses}. In his 1609 book \textsl{Astronomia Nova}
he formulated these laws:
\begin{enumerate}
\item
Each planet moves along an ellipse with the Sun at one focus.
\item
The vector from the Sun to the planet sweeps out equal
areas in equal times.
\item
The ratio of the squares of the periods of two planets is
equal to the ratio of the cubes of their semimajor axes.
\end{enumerate}
Perhaps even more importantly, we see in Kepler's work these
new features, listed by E.\ J.\ Dijksterhuis in his magnificent
book \textsl{The Mechanization of the World Picture: Pythagoras
to Newton}:
\begin{enumerate}
\item
Rejection of all arguments which are based solely on tradition
and authority.
\item
Independence of scientific inquiry of all philosophical and
theological tenets.
\item
Constant application of the mathematical mode of thought in the
formulation and elaboration of hypothesis.
\item
Rigorous verification of the results deduced by the latter by
means of an empiricism raised to the highest degree of accuracy
\end{enumerate}
\subsection*{Galileo Galilei}
In 1632 Galileo wrote a book on Copernican astronomy versus
Ptolemaic astronomy, \textsl{Dialogue Concerning the Two Chief World Systems}.
Among other things, he formulated the principle of relativity of motion
to explain why we wouldn't fall off a moving Earth:
\begin{quote}
Shut yourself up with some friend in the main cabin below decks on
some large ship, and have with you there some flies, butterflies, and
other small flying animals. Have a large bowl of water with some fish
in it; hang up a bottle that empties drop by drop into a wide vessel
beneath it. With the ship standing still, observe carefully how the
little animals fly with equal speed to all sides of the cabin. The
fish swim indifferently in all directions; the drops fall into the
vessel beneath; and, in throwing something to your friend, you need
throw it no more strongly in one direction than another, the distances
being equal; jumping with your feet together, you pass equal spaces in
every direction.
When you have observed all these things carefully (though doubtless
when the ship is standing still everything must happen in this way),
have the ship proceed with any speed you like, so long as the motion
is uniform and not fluctuating this way and that. You will discover
not the least change in all the effects named, nor could you tell from
any of them whether the ship was moving or standing still.
\end{quote}
He also argued that discounting wind resistance, a falling object
would fall at a constant acceleration independent of its mass.
Using the same geometrical argument as Oresme and others, he
saw that this meant it would fall a distance proportional to
$t^2$.
\subsection*{Isaac Newton}
Isaac Newton unified the work of Galileo and others on `terrestrial
mechanics' (falling bodies) with the work of Kepler and others on
`celestial mechanics' (the motion of planets). In his
\textsl{Philosophiae Naturalis Principia Mathematica},
published in 1687 after enormous delays, he formulated three
laws of motion:
\begin{itemize}
\item
\textit{Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi
uniformiter in directum, nisi quatenus a viribus impressis cogitur
statum illum mutare.}
Every body perseveres in its state of being at rest or of moving
uniformly straight forward, except insofar as it is compelled to
change its state by force impressed.
\item
\textit{Lex II: Mutationem motus proportionalem esse vi motrici impressae, et
fieri secundum lineam rectam qua vis illa imprimitur.}
The rate of change of momentum of a body is proportional to the
resultant force acting on the body and is in the same direction.
\item
\textit{Lex III: Actioni contrariam semper et fqualem esse reactionem: sive
corporum duorum actiones in se mutuo semper esse fquales et in partes
contrarias dirigi.}
All forces occur in pairs, and these two forces are equal in magnitude
and opposite in direction.
\end{itemize}
Note the third law is one of the doctrines condemned by the Bishop
of Paris.
Oversimplifying enormously, one can say that a key step here
was going from a first-order differential equation
(trying to explain velocity) to a second-order one (trying to
explain acceleration). Newton's second law can be formulated as a
differential equation
$$F = ma$$
or
$$F = m {d^2 q \over dt^2}$$
where $m$ is the body's mass, $q \maps \R \to \R^3$ is its position
as a function of time, and $F$ is the force upon it (typically some
function of $q$, ${dq\over dt}$, and perhaps $t$ as well. This
formulation is anachronistic since Newton didn't use \textit{vectors},
but he did invent the differential and integral calculus.
(So did Leibniz: Newton would write $\dot q$ while Leibniz wrote
$dq\over dt$.)
Newton's colleague Robert Hooke suggested had that the gravitational force
exerted by one body on another was inversely proportional to the
square of the distance between them. In an amazing \textit{tour de
force}, Newton was able to derive Kepler's three laws from this
assumption. In a guided homework exercise we will derive the first.
Then you'll see how smart Newton must have been!
\end{document}