The book attempts to explain the main features of celestial mechanics using a new and unique technique. Its emphasis in terms of applications is on the Solar System including its most peculiar properties (such as chaos resonances relativistic corrections etc.). All results are derived in a reasonably transparent manner so that anyone with a PC and a rudimentary knowledge of mathematics can readily verify them and even extend them to explore new situations if desired. The more mathematically oriented reader may also appreciate seeing quaternions as the basic algebraic tool of the new approach. Practicing astronomers may also be happy to learn that the method has eliminated the problem of zero divisors and does not require using the rather controversial averaging principle. The book is intended for astronomers and mathematicians interested in mechanical motion.