This book develops methods of computing astronomical phenomena from basic ideas. The position of a celestial body is defined by a vector, with components referred to a system of coordinate axes. The relations between various systems in regular use by astronomers are described. In cases where two systems differ in spatial orientation, they are related by a rotation matrix. These matrices are discussed in considerable detail in the mathematical notes. Other topics discussed include: Kepler's Laws and the dynamics of planetary motion, Precession and Nutation, transits of Venus and Mercury, Lagrange points.While no previous knowledge of Astronomy is necessary, it is assumed that the reader is familiar with elementary algebra, trigonometry and calculus.