This chapter, as the conclusion to Cohen's work attempts to show the connections of Newton to Galileo and Kepler, etc. In effect, Cohen asks us to see Newton as the great culmination of about 150 years of new work, conjecture, experiment, and methodology in the physical sciences.

In the absence of teaching assistants, please do not fret if you are uncomfortable with mathematics. If we had TAs, I would have them cover these basic mathematical relations that Cohen speaks of, but for now, let us concentrate on the larger meanings of chapter 7.

In particular, Newton's one great achievement is in adding the conception of forces to the scope of physics. Other scientists had spoken of forces, but only as things to be noted, not analyzed. Galileo did not speak of them; Kepler made up stuff about the sun sweeping the sun around; and people like Descartes and Huyghens knew they were important (in, for example, collisions), but could not effectively describe them or their actions.

There are only two concepts which I wold expect that you understand from this chapter:

• Inertia, both general and 'circular'

• The general outline of gravitation

The section Newtonian Anticipations (pp. 149-57) merely recounts the path of reasoning that Newton took from Halley's question of "what is the shape of the orbit..." to coming up with the basic formulation of F=ma. Of importance here, note that Cohen speaks in passing of the independence of perpendicular motions (see the photos in Plate VIII) -- basically that the fall of an object occurs the same whether it is moving horizontally or not. This is in fact a very important part of Newton which was indeed present in Galileo before, but Galileo never officially stated it as an axiom as did Newton.

Cohen's discussion about two cubes of aluminum (why aluminum, I wonder?) and how we decide which is heavier is instructive with regards to the concept of inertia. Think of when you heft something to tell how heavy it is. You usually come up with some sort of value, like 10 pounds or whatever. But how do you do it? If you have an item put into your hand and are not allowed to move, you will have a harder time guessing its weight. What we do is that we move it up and down. In effect, we are creating a velocity, or more specifically, an acceleration contrary to that of gravity (you 'weigh' the thing on the upswing, not the downswing). What we are effectively 'measuring' is some sense of the Force needed to accelerate it against gravity. Since, by experience, we know the amount of force needed to move various things the weight of which we know, we can compare this force to that we know and make an educated guess at the weight. In essence, we use F=ma: we know 'a' is the acceleration of gravity (although few of us know what that is numerically -- 9.81m/s², as it turns out), experientially and once we know the force, we can 'calculate' the mass from Newton's second law. Inertia, then, is the tendency of an object to not be moved by a force.

"The System of the World" (pp. 158-64) draws out the distinction between Newton the mathematician and Galileo the physicist. A number of things that Galileo would have found extremely objectionable, particularly the infinitude of the universe, Newton takes for granted. As a result, celestial mechanics furnishes Newton with a perfect example of mathematical inertia (as compared to physical inertia, which tends to decay because of friction). This section shows how it was that Newton realized that inertial motion is consistent with Kepler's 2^nd law of equal areas. The math on pp. 161-2 simply shows that instead of a body moving along a straight line making equal areas, if you add in the perpendicular force drawing the body towards a centre (a centripetal force and motion), then the equal area law will still hold.

The Masterstroke: Universal Gravitation (pp. 164-74) takes this 'proof' of Kelper's 2^nd law further, showing that Newton also noted Kepler's 3^rd law holds for Jupiter's moons (something Kepler, with poorer telescopes, could not have seen -- here you see that the advance made by Newton's reflecting telescope allowed advances in other areas). And if so, Newton reasoned, . That is, the Force that keeps things orbiting according to Kepler's laws is proportional to each of their masses and is also inversely proportional to the squares of their distances from one another.

The highly mathematical bits on pp. 165-72 we will just have to ignore for now. Those of you interested in it, enjoy, but if it frightens, pass over it with no fear that it will come back to haunt you.

Please only note on p. 170, Cohen in effect shows how Newton proved mathematically what Galileo had supposedly shown in dropping bodies from the Leaning Tower of Pisa: all bodies fall at the same rate, regardless of their mass. The point is, we get a convergence of older and newer ideas and methods which all point to the same (correct) conclusion.

1. since the Earth is a slightly squashed sphere (flatter at the poles), gravitational acceleration is a bit higher at the poles than at the equator (because you're farther from the centre of the Earth at the equator than you are at the poles)
2. why the earth wobbles as it spins, in effect accounting for the seasons exactly as he 'predicted'.
3. Why the ocean tides do what they do (gravitational attraction with the Moon, mainly)

The last section, pp. 180-84 is a very brief look at the later concepts of relativity which need not concern us here -- you might, however, come back to look at it when we do a bit on Einstein in the last week of class.