Sir William Rowan Hamilton
1805-1865
Irish Mathematician
Sir William Hamilton is perhaps best known for developing the theory of quaternions, the first non-cummutative form of algebra. However, he also made significant contributions to the theories of optics and other areas in mathematical physics.
Hamilton was born in Dublin, the son of an attorney. He showed signs of genius early, learning Latin, Greek, and Hebrew by the age of five. At age 13 he was introduced to mathematics and, at age 17, he brought an error in Pierre Laplace's book on celestial mechanics to the attention of John Brinkley, Ireland's Astronomer Royal. After this first meeting, Brinkley commented, "This young man, I do not say will be, but is, the first mathematician of his age."
During his undergraduate years, Hamilton was appointed professor of astronomy at Trinity College. This distinction was followed by his appointment as Astronomer Royal of Ireland. It turned out that both of these appointments were unsatisfying, as Hamilton's primary love was mathematics, not astronomy, and it was in mathematics that he did his most important work.
One of Hamilton's first major accomplishments was a theory of conical refraction, of interest to opticians and verified experimentally within months. He then began work in complexnumbers, systems of numbers in which one term is the square root of -1. Hamilton was among the first to treat complex numbers as points on a plane in which one coordinate is a real number and the other, an imaginary number. (An imaginary number is a number multiplied by i, the square root of -1, and is usually expressed as 2i, for example.)
Sir William Rowan Hamilton. (The Library of Congress. Reproduced by permission.)He also introduced the concept of the Hamiltonian, a way of describing the total energy of a system. In this system, the total energy H of a body is constant over any path it can take, making the dynamical properties of the body more easily and accurately described. In this same area of mathematical physics, Hamilton also described Hamilton's principle, worked on the Hamilton-Jacobi equations, and developed the Hamiltonian operator. All of these are considered developments of fundamental importance to the study of dynamical systems, and all are still taught to physicists today.
Hamilton spent many years working on quaternion theory in one form or another. This work was started by realizing that a description of the motion of an object in three dimensions requires a mathematical description in four dimensions. Such problems could not be handled by the mathematics of the day. After nearly a decade of work, Hamilton finally understood the solution to this problem in the form ofquaternions, which could be solved by using the first non-commutative algebra. An algebra is commutative if changing the order of the terms has no effect on the final solution. For example, 2 × 3 gives the same answer as 3 × 2. However, quaternions do not behave in this manner—they produce a different answer if the order of the terms is changed. Quaternions were not of much immediate use and, in fact, were not nearly as important as Hamilton had hoped. William Thomson (1824-1907), Lord Kelvin, wrote, "Quaternions came from Hamilton after his really good work had been done and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way." However, several decades later, Josiah Gibbs (1839-1903) was to find them exceptionally useful in developing vector calculus which, in turn, is widely used in physics today.
In marked contrast to his professional successes, Hamilton's personal life was a disappointment to him. In his early twenties he fell in love with a woman, but lost her to a more financially successful rival. After that, he seemed not to care much for any other women, and eventually married for the sake of marrying. He also had periodic problems with alcohol abuse, alternately controlling and losing control to his cravings. He died of a severe attack of gout at the age of 60, shortly after his election as the first foreign member of the National Academy of Sciences in the United States.
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