Until the beginning of the twentieth century, Newtonian mechanics
was thought to constitute a complete description of all types of
motion occurring in the universe. We now know that this is not the
case. The modern view is that Newton's model is only an approximation that is valid under certain circumstances.
The model breaks down when the velocities of the objects under
investigation approach the speed of light in a vacuum, and must be modified in accordance with Einstein's
*special theory of relativity*. The model also fails in regions of space that are sufficiently curved that the propositions of Euclidean geometry
do not hold to a good approximation, and must be augmented by Einstein's *general theory
of relativity*.
Finally, the model breaks down on atomic and subatomic lengthscales, and must be replaced by *quantum mechanics*.
In this book, we shall (almost entirely) neglect relativistic and quantum effects.
It follows that we must restrict our investigations to the motions of large (compared to an atom), slow (compared to the speed of light)
objects moving in Euclidean space. Fortunately, virtually all of the motions encountered in conventional celestial mechanics fall into
this category.

Newton very deliberately modeled his approach in the *Principia* on
that taken in Euclid's *Elements*.
Indeed, Newton's theory of motion has much in common with a conventional *axiomatic system*, such as Euclidean geometry. Like all axiomatic systems, Newtonian mechanics starts from a set of terms that are undefined within the
system. In this case, the fundamental terms are *mass*, *position*,
*time*, and *force*. It is taken for granted that we understand what these terms mean,
and, furthermore, that they correspond to measurable quantities that
can be ascribed to, or associated with, objects in the world
around us. In particular, it is assumed that the ideas of position
in space, distance in space, and position as a function of time in space, are
correctly described by conventional Euclidean vector algebra and vector calculus.
The next component
of an axiomatic system is a set of *axioms*; this is a set of
unproven propositions, involving the undefined terms, from which
all other propositions in the system can be derived via logic and mathematical
analysis. In the present case, the axioms are called *Newton's laws of
motion*, and can only be justified via experimental observation.
Note, incidentally, that Newton's laws, in their primitive form, are only applicable
to *point objects* (i.e., objects of negligible spatial extent). However, these laws can
be applied to extended objects by treating them as collections of point
objects.

One difference between an axiomatic system and a physical theory is that, in the latter case, even if a given prediction has been shown to follow necessarily from the axioms of the theory, it is still incumbent upon us to test the prediction against experimental observations. Lack of agreement might indicate faulty experimental data, faulty application of the theory (for instance, in the case of Newtonian mechanics, there might be forces at work that we have not identified), or, as a last resort, incorrectness of the theory. Fortunately, Newtonian mechanics has been found to give predictions that are in excellent agreement with experimental observations in all situations in which it would be expected to hold.

In the following, it is assumed that we know how to set up a rigid Cartesian frame of reference, and also how to measure the positions of point objects as functions of time within this frame. In addition, it is taken for granted that we have some basic familiarity with the laws of mechanics.