• Newton's Principia begins with a statement of his three laws of motions, and then proceeds methodically in deducing theorems and propositions from those axioms. This is certainly not an accurate portrayal of the order in which Newton discovered his laws (nor was it intended as such).
• At first sight, Newton's science looks like hypothetico-deductivism: Newton begins with his 3 laws of motion, deduces observational statements that are known to be true by observation, later adds his universal law of gravitation and deduces more observational facts, which increase the confirmation of his theory.
• This is not what is going on. There is plenty of deduction, but what it is not true observational statements that are being deduced, but false theoretical models (idealizations).
• If this is not what is going on then this is clearly important to the philosophy of science because Newton's Principia has been used as a measure of all science—any science that has not measured up to Newtonian standards has been burdened with the reputation of second rate science. So, if these standards are not what people have assumed, then this is extremely important to know.
• To understand what is going on in Newton's Principia, we need to look at some details.

• Under Galileo's idea of circular inertia, the moon could continue to move with uniform speed in a circle around the Earth until acted on by a force. On the other hand, the freely falling apple is subject to a force, so the moon and the apple have nothing much in common.
• Newton's first law of motion is a statement of linear inertia: It says that any body will continue to move in the same direction and the same speed until acted on by a force. Applied to the moon, it implies that the moon will continue to move in a straight line with uniform speed until acted on by a force. Because it does not so move, it is being acted on by a force; namely gravity. It has that much in common with the apple.
• The key step in Newton's solution was to conceive of acceleration as a change of the direction and speed (called velocity) rather then mere change of speed. Thus, the moon’s acceleration is due to its changing direction of motion, while the acceleration of the apple is due to its changing speed. Only after this new description of motion is introduced, could Newton argue that the observed rates of acceleration in both cases yielded estimates of the earth’s gravity that were in quantitative agreement with his inverse square law of gravitation.

This argument, and arguments like it, are what made Newton famous.

Derivation of Kepler's 2^nd law: Newton uses this result to show that (a) uniform linear motion obeys the area law, and (b) any uniform motion with an added motion (change in velocity, or acceleration) towards the center also obeys the area law. Part (b) covers the case of elliptical orbits, but is actually more general than that.

Derivation of Kepler's 1^st law: In Proposition XI, Problem VI, Book I, Newton proves that if the radius from a focus of an ellipse to a body on the ellipse sweeps out equal areas in equal times then the magnitude of the acceleration towards the focus is inversely proportional to the square of that distance. His also shows a converse to this theorem: that if the acceleration is inversely proportional to the square of that distance, then the path is either elliptical, parabolic or hyperbolic.

Derivation of Kepler's 3^rd law: He derives Kepler's third law by showing that if the planets are much lighter than the sun, then D³/T² is a measure of the mass of the sun, and is therefore constant.

• Thus, there is a sense in which Newton has derived Kepler's three laws of motion in a way that also applies to the motion of the moons around Jupiter and our moon around the Earth.
• But is this anything to be proud of? Aren't Kepler's laws false, in which case the derivations simply refute the theory! So, what has Newton achieved, and how should it be understood?

• Newton has derived a model for the motion of planets around the sun that is observationally equivalent to Kepler's model of planetary motion. This was especially important because Kepler's work was not highly regarded by more traditional Copernicans (like Galileo) because much of it was steeped in mysticism, and it departed from 'pure' circular motion (which is why Copernicus has rejected Ptolemy's equant construction in the first place).
• The derivation used a number of auxiliary assumptions that he knew to be false. Newton's Keplerian model was an idealization. If it showed anything, it showed that Newton was using false premises.
• The auxiliary assumption included the assumption that the sun and the planets are uniformly spherical (and can therefore be treated as point masses), and that the net force on a planet is towards the sun (so the attraction of other planets is negligible).
• Nevertheless, the derivation is important because the consequences of the idealization fit the observations fairly well, and it was plausible to suppose that any inaccuracies could be blamed on the inaccuracies of the auxiliary assumptions.
• Unlike Kepler's theory, Newton provided a recipe for constructing new models.
• Unlike Copernicus's program, it promised simpler and more unified models, because the masses of celestial bodies (plus the universal constant of gravitation G) and initial positions and velocities were the only adjustable parameters needed. So, unification and simplicity were key advantages over Copernican program, which certainly did not lack the potential for expansion.
• To use Lakatos's term, Newton had defined a promising scientific research program.
• Newton's deductions showed that he could do everything his predecessors could do, plus some more.
• He provided a recipe for constructing new models of planetary with few new adjustable parameters. (For example, he needed to add a parameter for the oblateness of the Earth to account for the precession of equinoxes.)
• He also developed this program to prove that its promise was real.

According to the myth, Newton thought of the idea of gravity when an apple fell on his head as he was gazing at the moon in the day-time sky. While the story is untrue, there is an important grain of truth behind it. For it suggests that the key element behind Newton’s theory is the unification of celestial and terrestrial motion. Note that unification is not mentioned in the hypothetico-deductivist story.

How did Newton account for Galileo’s law of projectile motion? Newton’s analysis comes in two stages.

• In Section XII, Book I, Newton calculates the attractive forces of spherical bodies like the Earth. There he shows that any solid spherical body with a spherically symmetric mass distribution will act on bodies on or above its surface as if all the mass were concentrated in a single point at the center. The projectile may be as close to the surface of the sphere as you like—there is no approximation involved. Intuitively, the result can be understood in the following way. Every part of the Earth will attract a given object on or above its surface. But for every piece of the Earth off center, there is a corresponding piece off center on the other side at the same distance, and their combined attraction is towards the center of the Earth. This result allows Newton to treat the Earth and other celestial bodies as if they were point masses. Of course, this is merely a convenient fiction. There is no need to interpret Newton as employing the absurd auxiliary assumption that such bodies are point masses. The only idealization is that celestial bodies are spherically symmetric in the way their mass is distributed from their center.
• Newton’s next step is to treat projectile motion as a small truncated segment of a Keplerian orbit around the Earth’s center. Because of the relatively large distance to the center of the Earth, the radius drawn from the Earth’s center to the projectile is approximately constant in length throughout the motion. So, the radius moves at a steady angular speed in order to sweep out equal areas in equal times. The successive positions of this radius are approximately parallel when viewed on the surface of the Earth, so the area law accounts for the horizontal component of the motion. The vertical component of the motion is an acceleration towards the center of the Earth proportional to M_e/R 2, where M_e is the mass of the Earth and R is the distance from the center of the Earth to the projectile. Given that R does not change appreciably during the motion, the acceleration is constant, and this is Galileo’s law.

What is the relationship amongst N, K, and G?

We could ask why N is better than K or G, but this has an easy answer—N has broader scope than G and N has broader scope than K. The correct question is ask is why N is better than a combination of K and G, which we denote by K & G.

• As Newton argued, N is a model of Newtonian mechanics, and N implies K and N implies K.
• However, the converse does not hold. K & G is not strong enough to entail N. Why not? Because N implies that the acceleration of the moon and the acceleration of the apple is proportional to the mass of the Earth, m, and inversely proportional to the respective squared distance from the center of the Earth. If these distances are known, then we obtain two independent measurement of the Earth's mass.
• N says that these measurements should agree. The K & G coalition does not require that these measurement agree. Therefore N is strictly stronger than K & G.

There are at least four ways of describing the virtue of N over K & G.

1. N is more falsifiable than K & G.
2. K & G can accommodate any observational facts that N can accommodate, but there are some predictions that N could make that K & G could not. For example, Given data about terrestrial motion, we can infer the mass of the Earth and use this information to predict the acceleration of the moon (given its distance from the Earth).
3. N explains the agreement in the independent measurements by saying that they arise from a common cause (namely, the Earth's gravity). K & G dismisses this as a mere coincidence.
4. N has fewer adjustable parameters than K & G.

Therefore, N does not do "just as well" as K & G, it does better. And the model N was generated by scientific research program that has a strong positive heuristic (Lakatos's term)—namely a recipe for constructing new models that involve minimal complications of the previous ones.

• This is same pattern that we saw in favor of Copernicus over Ptolemy.
• The deductions in Newton's Principia are of models from the theory plus auxiliary assumptions.
• The models fit the observations to a certain degree. They are not required, nor are they expected, to fit perfectly.

1. Hypothetico-deductivism assumes that observational statements are derived from theories. However, Kepler's laws, or Galileo's laws are not observation statements.
2. Even when observation statements are derived, fit is not measured by the truth of predictions, but by their degree of fit. It is a statistical degree of fit and not one of logic (e.g., whether two independent measurements agree).
3. The problem of idealization: Popper's falsificationism (being a kind of hypothetico-deductivism) implies that idealizations should be abandoned. However, idealizations are not abandoned if they provide sufficiently good predictions. This suggests that it is good prediction, rather than true models, that is the goal that scientists pursues.
4. It does not mention, or place any significance in, things like simplicity or unification.