A Priori
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teaching purposes.
An a priori proof is one that does not require
scientific observations. It is based on reason,
logic, and mathematics. An a priori proof differs
from an empirical proof which relies on
experiments or observations.
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Newton and Kepler were searching for an a priori explanation of
orbits. They were keenly interested in determining the underlying
reasons for the planetary laws.
The traditional approach to proving Kepler's Planetary Laws is
not a priori since it relies on one of two premises. Either we
assume that Newton's Inverse Square Law of Gravitational Force
is true or we assume that Kepler's observation that orbits are
ellipses is true. A proof can only be truly a priori if it does not rely
on one of these premises.
Accordingly, in non a priori fashion, Newton demonstrated that if
it is given that orbits are elliptical, then the force must vary
inversely with the square of the distance. His demonstration,
based on the empirical observation that orbits are elliptical, is the
basis for the traditional, non a priori, explanation of orbits. In fact,
most modern texts on celestial mechanics begin by using
Newton's Universal Law of Gravitation as the premise for their
proofs.
But there is a way to prove the planetary laws without relying on
astronomical observations. The a priori proof presented in Orbits
Explained begins with Newton's demonstration that equal areas
are swept in equal times as a planet moves past the Sun,
regardless of the relationship between distance and gravitational
force. The a priori proof continues by employing a new
mathematical device, the hododyne, to generate inverse
proportions. Conveniently, the properties of tangential velocity
and distance are inversely proportional to each other - as is
easily demonstrated geometrically according to the concept of
equal areas swept in equal times. By assigning these properties
to the hododyne we can generate a diagram which exactly
determines the position and velocity of the planet in a priori
fashion. Many fundamental properties of planetary motion
including the elliptical nature of orbits unfold as we study the
hododyne and the orbits that it generates.