Newton went beyond his simple laws of motion and gravitation to develop a whole set of mathematics to describe and calculate orbits. Today we call this mathematics calculus. The key to calculus is the use of vectors. A vector is a quantity that has both magnitude and direction. It is typically represented symbolically by an arrow in the proper direction, whose length is proportional to the magnitude of the vector. Although a vector has magnitude and direction, it does not have a position. A vector is not altered if it is displaced parallel to itself as long as its length is not changed.

Because vectors are different from ordinary (i.e., scalar) quantities, all mathematical operations involving vectors must be carefully defined. If vector ‘A’ is added to vector ‘B’, the result is another vector,’C’, written A + B = C. The operation is performed by displacing ‘B’ so that it begins where ‘A’ ends. ‘C’ is then the vector that starts where ‘A’ begins and ends where ‘B’ ends.

Vector subtraction is defined by A – B = A + (-B), where the vector ‘-B’ has the same magnitude as ‘B’ but the opposition direction. A vector may be multiplied by a scalar. Thus, for example, the vector ‘2A’ has the same direction as ‘A’ but is twice as long.

Newton applied vectors in terms of force. A body is added on by a vector force as shown above. Forces can be added just like vectors, so that force ‘1’ and force ‘2’ add together to produce the total force, ‘F’. Total force ‘F’ can also be broken into components ‘x’ and ‘y’ that correspond to the forces in the ‘x’ and ‘y’ directions (for example, along a road and with gravity).

A particle moving with constant velocity ‘v’ suffers a displacement ‘s’ in time ‘t’ given by s = vt. The vector ‘v’ has been multiplied by the scalar ‘t’ to give a new vector, ‘s’, which has the same direction as ‘v’ but cannot be compared to ‘v’ in magnitude (a displacement of one meter is neither bigger nor smaller than a velocity of one meter per second). This is a typical example of a phenomenon that might be represented by different equations in differently oriented Cartesian coordinate systems but that has a single vector equation (for all observers not moving with respect to one another).

For a particle of mass ‘m’, a force is applied with results in an acceleration ‘a’. The acceleration changes the velocity vector by a small amount, ‘delta v’, every interval of time, ‘delta t’. This results in trajectories, a vector map of the changes in position from an origin, the vector ‘x’ and the velocities, vector ‘v’.

With vector calculus, Newton was able to develop a cosmology which included the underlying cause of planetary motion, gravity, completed the solar system model begun by the Babylonians and early Greeks. The mathematical formulation of Newton’s dynamic model of the solar system became the science of celestial mechanics, the greatest of the deterministic sciences.

Although Newtonian mechanics was the grand achievement of the 1700’s, it was by no means the final answer. For example, the equations of orbits could be solved for two bodies, but could not be solved for three or more bodies. The three body problem puzzled astronomers for years until it was learned that some mathematical problems suffer from deterministic chaos, where dynamical systems have apparently random or unpredictable behavior.

- kinematic description of the Solar System (Kepler)
- dynamical description of the Solar System (Newton)
- development of celestial mechanics
- 1650’s to 1700’s = improvements in telescope technology = more precise measurements of planetary positions = more accurate tests to Newton’s theory of gravity
- 1780: Herschel accidentally discovers Uranus (new planet to continue to test theory of gravity)
- 1845: perturbations in Uranus’ orbit used to predict the position of a new planet by Adam/Leverrier (Neptune) = crowning achievement for celestial mechanics