Sir William Rowan Hamilton (1805-65) was a distinguished Irish mathematician who worked in the fields of classical mechanics, optics and algebra, as well as in physics and astronomy. Hamilton was the discoverer of quaternions, which are defined as a non-commutative number system which extends the complex numbers. He first described them in 1843, and devoted much of his subsequent life to studying and lecturing on the concept. This book was published posthumously in 1866, with the final editing by his son. Until they were replaced, from the mid-1880s, by vector analysis, quaternions were taught as a major topic in advanced mathematics at most universities, and their utility in describing spatial relations has led to a revival of interest in them since the late twentieth century.