Let me give the most bare
explanation of the diagram. Consider the somewhat
darker line running from A to B to C to D to
E to F. This will represent the path of a body
in motion from A to F.
Consider a body at A. If it is in motion and
unaffected by anything else, it will move in
a straight line-- say, towards B. AB thus represents
its motion under its own impetus.
Now consider point S as a center of force–like
a magnet, or a large body exerting a gravitational
force on another. Newton says: imagine that
the body at A is acted on by S in such away
that it is pulled toward S–again, the
way that a magnet or body with gravitational
force would. Now, the body at A is impelled
by its own motion to move in a straight line
from A to B to c. If there were no body affecting
it at S, it would just keep going on that straight
line from A to B to c until–- until it
ran into something, or some other force affected
it . . . .
Now consider the effect of S on this body. Let
us say that the effect of S acts instantaneously
on the body when it hits B to draw it in from
B to V. Then it would move from V to C, just
as it would have from B to c–VC parallel
to Bc and equal in length to it. This allows
us to see C as the position that the body will
actually take under the joint effect of its
own motion and the force at S. The same will
go for d and D etc–in each case, the miniscule
letter shows where the body would have been
if there had been no force at S, and the majuscule
shows where it actually ends up.
Lastly, imagine that instead of six steps to
get from A to F, we consider 100 steps-- or
a thousand. With each increase in the number
of steps, the distance from one "stop"
to another shrinks. Newton asks us to consider
the motion (and the resulting path) when we
shrink the distance between the stops down to
nothing-- "infinitesimally small."
The path gets closer and closer to a curve as
these increments shrink.
Newton is simply asking us
to imagine the orbit of a planet (or an electron)
and helping us to understand what it means that
that orbital motion could be due to the joint
operation of two forces–a straight line
force within the moving body, and another straight
line force, acting on it at every instant, pulling
it in toward itself.
The wonder of the proposition
is not that Newton somehow demonstrates that
planetary motion can be decomposed into two
forces–that is not what the proposition
demonstrates but what it assumes. The wonder
of the proposition is how much he makes knowable
about that motion. And he does it all by a kind
of geometrical reasoning.
The proposition is dear to
me and emblematic of the project for several
reasons. The proposition and its diagram have
a surface simplicity (so few points and lines,
such elegant geometrical reasoning) yet it is
impossible for me to master it. No matter how
many times I return to it, it remains not only
marvelous but nearly opaque. Winemaking has
the same kind of marvelous opacity to me: one
has glimpses into what is going on, one has
moments of understanding; but what is going
on is so immeasurably deep and complex that
the glimpses remain punctuations at best. But
it is nonetheless possible to follow Newton,
even if it is impossible for me to master him.
The same is true with the wine: there is something
luminous and knowable going on, even if it is
impossible to master it.
It is dear for another important
reason too. When I was learning how to make
wine, the son of my teacher and mentor was in
the last few years of his home schooling. His
academy was in need of a new faculty member
to enliven the curriculum. John Kongsgaard proposed
that as he was teaching me, I would teach his
and Maggy's son Alex. It was a wonderful idea.
Alex and I met two to three times a week, for
several hours at a time, some of them on the
Vallejo-San Francisco ferry (carrying me back
from work), some of them at my kitchen table.
We did this for two school years, and studied
everything from Beowulf to Genesis to Dashiell
Hammett to Ancient Greek. But somehow at the
center of everything for me was Newton. We spent
a whole year studying Newton together, drawing
obscure diagrams on the tables on the ferry,
and the culmination of our whole year's study
was being able to demonstrate this diagram,
Proposition 1 of the Principia. Alex's final
exam was a public demonstration before his parents
on an easel in my kitchen, followed by dinner.
Very soon, Alex was not only my student, but
my friend, my helper in winemaking, and my teacher