Principia Book 1 Section 3

At some point I gave up on trying to get the details of each step of each proof and instead just figure out the chain of ideas from one proposition to the next.  Section 2 went from fundamental results that I can articulate to technical results whose motivation was hard to follow at the time.  In fact, even when I think I have unpacked them I then find myself wondering if I am missing the point.  This is not a book that I will be able to digest, only tour. But in section 3 he uses these results to conclude that conic orbits with the sun at the center  imply that objects are attracted to the sun via an inverse square power of the distance from the sun, and that the inverse square law implies Kepler's laws.  That alone would guarantee his immortality, but he still has more to do.  Let's see what he has for us next.

Principia: Book 1, section 2, Proposition 4

This one was hard, because I'm pretty sure there's an error in the transcription.  The proof says that the proposition can be derived from Lemma 7 of the previous section, but I see no obvious way to get it from that.  OTOH, I can easily see how to get it from Lemma 11.

The first 7 corollaries are great.  The guy derives the inverse square law of gravity for circular orbits, assuming Kepler's law (T^2 ~ R^3).  Awesome.

Then Corollary 9 threw me through a loop.  The translators used the term "mean proportional" rather than "geometric mean."  But I sorted it out.

This book is a hard slog, but worth it.

Principia Book 1, Section 2, Propositions 1-3: Tricky but rewarding

The Principia is almost entirely geometry, not algebra, but it is worth it.  The first proposition in Book 1, Section 2 has the most intuitive diagram ever drawn for explaining centripetal forces.  The proofs for the theorem and corollaries derived from it are a bit of a slog for somebody who hasn't done tons of geometric proofs in forever, but the basic insight into centripetal motion from the diagram itself is perfect.  Also, this very first proposition is equal areas in equal times.  The man does not waste any time, he just gets right into fundamental results, and then his corollaries establish visual interpretations of forces.  Beautiful stuff.

In the second proposition he just shows that if you observe something moving in such a way as to sweep out equal areas in equal times (looking at arcs drawn about some designated center) then it must be acting under a centripetal force.

In the third proposition he argues that if a body is tracing out equal areas in equal times with respect to some accelerating object, then the first body must be under the combined influence of a centripetal force from the second object and also the same accelerative force as the second object is feeling.  In other words, Newton translates his results to non-inertial frames.

So far the man has built laid down basic laws of motion, described a painstaking experiment, built up calculus, derived one of Kepler's Laws, and developed some physics for non-inertial frames.  This is why physicists regard him as something of a deity.  He's blending together a bunch of techniques and fundamental concepts, any one of which could easily be a claim to some fame in the history of physics, and at the same time he's blending together skill sets that few physicists will ever master simultaneously.

This is why we love him.

Principia, Book 1, Section 1

This section is entirely geometric proofs of lemmas concerning either areas or ratios of lengths or areas in limiting cases as angles or lengths approach zero, or as the number of divisions becomes infinite.  It's a hard slog, and I have to admit that I mostly read to get a sense of Newton's style rather than the details.  I am proud of myself, though, for being able to deduce that Lemma 10 is a proof that dx=0.5*a(0)*dt^2 for small dt and v(0)=0 and that Lemma 11 is a proof that y~theta^2 for a pendulum set up such that y=0 when theta=0.

Section 2 is physics rather than geometry; hopefully it will be an easier read.

Newton: "Axioms, or Laws of Motion"

After stating the three laws of motion, Newton starts into corollaries.  The first two corollaries involve the parallelogram law of forces, resolution of forces into components, and an example with tension in ropes.  The next two corollaries (3-5) are more interesting, involving conservation of momentum (though not stated in that language; to Newton momentum was "quantity of motion") and the conservation of center of mass velocity.

Corollaries 5 and 6 show that the relative motion of a system of interacting bodies is unaffected if we move to a different inertial reference frame, or to a non-inertial reference frame at constant acceleration.

The Scholium shows us what a genius he is.  He cites prior work by Huygens, Wren, and Wallis on collisions, but notes that they only worked out elastic collisions.  He works out an example with an inelastic collision of two hard objects on pendula, shows how to estimate the effect of dissipative forces in a self-consistent manner, and then describes an experiment that he did to test his calculations (involving a 10 foot pendulum and errors no larger than 3 inches for the maximum height reached by objects, i.e. 2.5% error).

Having showed his skill as an experimental, he uses symmetry arguments and the impossibility of perpetual motion to demonstrate the validity of the Third Law for attractive forces, including a thought experiment on the stability of the earth as a self-gravitating object.  Since the Third Law can be shown (in the Lagrangian formalism) to arise from translational invariance, I like his use of symmetry arguments here.  It may be the earliest precursor of Noether's Theorem.

Newton: Definitions

I just finished reading the first section of the Principia, on definitions.  In discussions of Definitions 3 ("Inherent force of matter") and 4 ("Impressed force") Newton repeatedly refers to the "force of inertia."  Given that a person pushing on an object to change its velocity feels a force from the object when they make contact (the normal force, equal and opposite to the impressed force) I suppose that one can forgive him for this conceptual error.  I am amused by the thought of modern physics pedagogues (whether old school traditionalists lecturing kids about how inertia isn't a force or hip and modern interactive instructors giving kids group activities on the top) scolding him.

Then he talks about the difference between absolute motion and relative motion, absolute motion being motion relative to space and relative motion being relative to another object.  He makes the important point that you cannot discern absolute motion by looking at the relative motion of two different objects because you don't know the absolute motion of either.  However, he then explores his famous bucket thought experiment to argue that there are some cases in which absolute acceleration can be inferred.  I've already discussed that thought experiment and Mach's reaction to it, so I have nothing to add here.  I will just say that the real significance of the Principia is not his conceptual understanding of physics (a modern pedagogue could easily find much to scold him for) but rather his ability to add new ideas to a big, bold, framework and then apply those ideas unflinchingly and with no reluctance to generalize, and extract predictions for the motion of the planets.

Next up:  His axioms, or laws of motion.

More sages on the pages

The Atlantic has an article defending the college lecture, written by Christine Gross-Loh.  It is interesting to see this; they are hardly a counter-establishment venue, but they do tend to publish a certain amount of commentary deemed to be within the range of safe contrarianism.

I learned of this article from a piece by Joshua Kim at Inside Higher Ed.  Kim argues that most people in instructional design hold opinions on lecturing that are more nuanced than those that the Atlantic article is responding to.  That may be true, but the currents that Gross-Loh highlights are nonetheless real.  I've submitted a comment saying that; I won't regurgitate it here because it's nothing I haven't said a million times before and it will show up soon enough at IHE.