Fermat's Last Theorem: Sir William Rowan Hamilton

Thursday, October 02, 2008

Sir William Rowan Hamilton

William Rowan Hamilton was born on August 4, 1805 in Dublin, Ireland. He was the fourth of nine children. Young William showed tremendous intellectual ability very early. By age 5, he had already learned Latin, Greek, and Hebrew.

When William was 12, he heard about an American Zerah Colburn who could do impressive calculations all in his head. Hamilton entered a mathematical competition where Zerah had also entered. His loss to Colburn drove him to study mathematics.

At 13, he began to study on his own a classic text on algebra. At 15, he started reading the works of Sir Isaac Newton and Pierre-Simon Laplace. Then, when 18, he started to attract mathematical attention when he found an error in one of Laplace's calculations.

In 1823, he began attending Trinity College. His first year he received an honor (an "optime") in the Classics that is bestowed on students only once every 20 years. Two years later, he received an "optime" in physiccs. For his senior thesis, he wrote an essay where he introducted a function that characterized optics. His outstanding performance at Trinity so astounded the faculty, that the age of 22, before he graduated, he was offered a professorship in astronomy at the college. His title was now Royal Astronomer of Ireland.

Before starting his new position, Hamilton decided to travel to England and Scotland. As part of this trip, he met the poet William Wordsworth. They quickly became friends. While the Royal Astronomer, Hamilton chose to focus on mathematics. He expanded on his senior thesis on a characteristic function in optics. Hamilton showed that the optical properties of a system could be derived from this characteristic function and its partial derivatives.

In 1832, Hamilton built on the theories of Augustin Fresnel and Thomas Young to make predictions about "internal" and "external" conical refraction of light through a crystal. These predictions created quite a stir when they were verified by Humphrey Lloyd through experiment. This was seen as a real triumph of mathematical science.

Between 1834 and 1835, Hamilton published two very important papers on dynamics. Hamilton had found that like optics, dynamics could also be described by a characteristic function. Hamilton introduced what is today called the "Hamiltonian" which is a mathematical form that characterizes the motion of a conservative dynamic system.

It was well known since the time of Carl Friedrich Gauss that the set of complex numbers could be mapped to a 2D coordinate system. Hamilton believed that it should be possible to find a number system that corresponded to a 3D coordinate system. In working through this problem in 1843, Hamilton came up with the concept of "quaternions." This he would consider his greatest achievement. Quaternions was one of the first non-commutative algebras. It is a four-dimensional complex number. In defining quaternion, Hamilton introduced the ideas of scalars and vectors.

When James Clerk Maxwell published his groundbreaking Treatise on Electricity and Magnetism that established the scientific relationship between electricity and magnetism, he used the mathematics of quaternions to state his fundamental equations.

Maxwell's theory of optics initially did not a significant impact on the state of the art. Later, towards the latter part of the nineteenth century, there was a resurgence in their influence and today are used in the design of optical instruments.

Maxwel's theory of dynamics from the beginning was extremely influential in the development of physics. The underyling model was greatly expanded by Carl Jacobi and Hamilton's theory of dynamics played a very influential role in the development of quantum mechanics.

Quaternions was later superceded by the development of modern vector algebra which was developed by Josiah Willard Gibbs and Oliver Heaviside. They created vector algebra as part of an effort to simplify quaternions and to make the subject more intuitive.

Hamilton married in 1833 and had three children, two boys and one girl. He was knighted in 1835. In 1837, he was elected president of the Royal Irish Academy. He held this position until 1846.

Towards the end of his life, Hamilton focused completely on quaternions and continued to extend the length of his treatise on it. He overspent the Trinity funds that were available for this work and his family began to experience financial difficulties. It was during these difficult times that Hamilton grew very sick. He died on September 2, 1865 at the age of 60.

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