Jan. 12 Reading Guide

Reading Guide ------ 12 January 2000

Isaac Newton and the Principia and

Henry Cavendish on the Gravitational Constant

Newton's Principia

This series of excerpts from the Principia are designed to give you a sense of the range of topics Newton addressed in the book, as well as how he addressed them. This first thing to notice is that in places he seems to ramble, as if he was working the concepts out as he went along. This may be partially true, but it more so accounted for by two other things: the general style of prose writing in the late 17^th century which tended to preserve the thought processes that lead the scientists (or other philosophers) to their conclusions and the fact that the audience for his book (which was admittedly small) would have indeed needed to have been lead along by the hand. Remember, at this time there was not coherent body of physics instruction, least of all mathematical physics. There were certainly no textbooks in the modern sense. Newton was (as I have repeatedly argued) actually summarizing the movements which lead up to him, but he was in one sense breaking new ground for the Principia itself was designed to be a self-contained physics textbook wherein the student could learn the entire framework of mechanics in one go, albeit a rather long go.

The selection for your reading here can be thought of as 6 distinct sections; each one for us has a different meaning and importance. They are

The introductory preface, setting out where the Principia 'fits' within the intellectual world in which it was written (pp. 145-7)
The initial definitions about ideas of matter and motion (note the Cartesian influence) to be used in the book (pp. 147-50)
The Scholium (the equivalent of a footnote) on the difference of absolute and relative things (pp. 150-55)
His three axioms or laws of motion (pp. 155-56)
The "Rules of Reasoning in Philosophy" by which he really means "Reasoning in [Natural] Philosophy", or Science (pp. 156-58) and
The "General Scholium" or conclusion of the entire book (added in the 1713 2^nd edition) which explains where he (by this time) believed the importance of his work to lie (pp. 157-61)

Let us take up each extract individually:

The first section, which is the opening preface to the Principia makes two important distinctions (or rather, connections): mechanics the people and mechanics the science are intimately related, and second, that the science of mechanics (or the natural philosophy of mechanics, or the mechanical philosophy) is, in its essence, reducible to geometry. The first connection is interesting because the Principia is written for anyone but mechanics (that is, common laborers), and yet Newton points out that in fact it is dealing with the very same thing, albeit at a considerably more refined and in effect perfect manner. Here, then, is the connection with the Baconian tradition immediately established. It would be another 50 or 100 years before the concept of teaching mechanics to mechanics (that is, the science to the people) would or could be seriously entertained, but it is clear that Newton conceived of his great work to be a step in that direction. Ultimately, the last 9 lines on p. 146 ("I wish we could derive... method of philosophy") are the goal of his work. Please note them well.

On the matter of geometry, the bulk of p. 146 sets out this importance. As mentioned in class, Newton developed calculus to develop his ideas, but could not write the results in this new "language" because no one else in the world yet understood it. Therefore, he wrote the entire Principia in sometimes very elaborate geometric fashion, and all throughout he is mainly relating line lengths to line lengths, angles to angles, and areas to areas. At the end of the first section, note that he does fully admit that the work was inspired and driven by Edmond Halley.

Section two, on his definitions, is straightforward. There are 5 definitions (and the modern equivalents are noted in italics in the relation):

quantity of matter (=volume x density = mass)

quantity of motion (=quantity of matter x velocity = mass x velocity = momentum)

"vis insita" (or "innate force") which gets equated with "vis intertia" (or "force of inactivity - inertia)

impressed force -- that which causes a change in motion (force, generally)

centripetal force -- that which pushes or pulls a body towards a point (force, in the case of circular motion)

There are a number of small things to notice within these definitions, other than knowing what they are.

First, in Def. I, Newton says "I have no regard in this place to a medium, if any such there is, that freely pervades the intersticies between the parts of bodies." This is a direct attack on Cartesian matter theory (recall the 3 sizes of particles, the smallest of which fill the voids between the larger). This "medium" is what I have called in class the "aether", a word and name worth remembering for later. In very many ways, the Principia is a direct attack on Cartesian everything.
The key to understanding inertia in Def. III is to understand that it is the "thing" that opposes forces, and everything in Newtonian mechanics is about forces. Thus, inertia is the thing that all of Newtonian mechanics is working against. And, please understand that inertia is not only the thing that keeps bodies at rest; it is also what keeps them in uniform motion.
Keep in mind that there are 2 kinds of general force in Def. IV: forces of percussion (i.e., sudden shocks) and forces of pressure (steady and continuous forces). Today we would say that both are the same kind of force and differ only in the amount of time each is applied, percussion being a larger (usually) force acting over a ver short time (like a baseball hit) and pressure being a more modes (usually) force applied over a long time or continually. Both can have the same effect -- enough small pressure over a long enough time (for example, water flowing through the Grand Canyon) can have the same effect as a very large force happening instantaneously (a whack-load of dynamite trying to blast a new Grand Canyon).
In Def. V, on centripetal forces, note that here is where Newton envisions an orbit as "falling around" a body: the orbiting object is moving fast enough that as it falls, it continually misses the Earth. Also note that in discussing why the Moon goes round the Earth, he still hedges his bets (since he can't prove it), saying, "if it [the moon] is endued with gravity or any other force."

The "Scholium" or "lesson" on absolute and relative time, space, and motion, is exceedingly important if we were eventually going to talk about Albert Einstein's theory of relativity at any great length. Since we are (sadly) not going to have time to cover that in anything more than topical detail, the important image here to keep in mind is the sailor walking towards the stern of the ship as the ship sails forward. To the sailor, he seems to be walking 1 kph towards the stern. To the observer on the quayside, he looks like he is moving at 9 kph in the same direction of the ship. And of course, since the earth is spinning he (and the observer) are actually moving eastward (regardless of the ships direction of ravel) at a much higher speed. For Newton, the key is that you can tell which motions are relative (e.g., the sailor with respect to the ship) by measuring the forces between them (p. 153). FORCES become the determining factor, not the velocities. Nonetheless, Newton realizes that relative motions are useful: "And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical discussions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them" (p. 152), although he certainly realizes that doing so is difficult (p. 154).

The laws of motion, I have dealt with at some length in class. It is required that you are able to recite them,

Regarding the rules of reasoning in philosophy, these four rules do effectively define "Newtonian Science", a form of science which we still largely follow today. Here Newton's discussion of causes verges on the clearly philosophical part of "natural philosophy" and in Rules I and II he defines a stripped down (what we may call common-sense) form of interpretation where the simplest solution is correct. (You may recall the narrator of the Galileo spoof noted that that is the rule for science: the simplest solution is always the best).

Rule II speaks to questions of matter. Elsewhere in the Principia Newton tends to try to ignore question of the material of nature, but here he sets out a rule that once again undermines the Cartesian way of doing things. In essence, he is arguing that it is useless to hypothesize about things which we cannot measure/determine by means of some sort of physical experiment. Unfortunately, here Newton is not as his best. Baigrie quite correctly notes that modern physics would have trouble with Newton's statement about hardness (with respect to phase changes), but there is also the question of what an experiment can demonstrate to our senses. As we have discussed before, instruments can often extend the senses to points where the philosophical questions of whether we are "sensing" reality can become relevant. That notwithstanding, notice finally that Newton here makes his "conclusion" of the reality of gravity seem as a fundamental supposition for what is to come.

And finally, Rule IV privileges induction (rather than deduction) in the study of science. That is, the search for general principles from specific instances, rather than the other way around.

The final section of this reading, and in fact, of the entire Principia, is the "General Scholium" (GS) The word scholium means "a small lecture or debate" and this "General" scholium was added by Newton to the second edition of the Principia as a way of partially answering some of the critics, but also as a way of extending the importance of his findings. Most scientists may not have taken the program of the GS too seriously, but many people in society certainly did do so.

For the purposes of the course, the GC does two things. First it sums up the general findings of the Principia, which I shall leave it to you to take note of, but beyond that, it demonstrates how important theology was to Newton (and may contemporary scientists) in terms of giving the ultimate explanations for the science. It may be true that Newton believed that gravity "caused" the planets to orbit as they do, but the ultimate question of "why gravity?" must (in Newton's opinion) be answered by recourse to God. I mention this not to particularly support, defend, or endorse Christian theology, but to specifically and once again point out that science and religion are NOT at odds in this case. In fact, they are seen as mutually supporting and reinforcing one another.

Cavendish on the Gravitational Constant

In jumping ahead just about 100 years from the Principia, in Cavindish's paper, we can see the outcome of Newton's "grand synthesis" in the attempt to actually quantify the gravitational force between two bodies. Newton had determined that the force of gravity is defined by F=(GMm)/(r²). As astronomers observed the heavens and measured the motions of the planets they could quite easily determine 'r', the distances between them and from their motions could figure out what the forces should be. From the known size of the Earth, they could then calculate its volume and make a guess at 'm', its mass. So, from Newton's equation, they could calculate the only unknown(s), 'GM', but had no way of separating the two. What Cavendish proposed to do was to try to measure 'GM' in the laboratory by using two balls of lead, where he did in fact know the 'm' and the 'M'. Then, 'G' could be calculates, but only because by this time scientists had agreed that gravitation does in fact bind the universe together, and that all bodies -- terrestrial or celestial -- do attract one another through gravitation.

Beyond the general understanding that Newton's Principia did in fact bear fruit in the laboratory, the only other thing I would ask you to take note of in this reading is how elaborate the physical apparatus had become by the end of the 18^th century and how sensitive the scientists were to outside sources of error such as wind, light, and distortions in the apparatus itself.

When studying this week's readings, please spend more time understanding Newton.