# William Rowan Hamilton

William Rowan Hamilton

Sir William Rowan Hamilton (born August 4, 1805 in Dublin , † September 2, 1865 in Dunsink near Dublin) was an Irish mathematician and physicist who is best known for his contributions to mechanics and for his introduction and study of quaternions .

Hamilton studied mathematics in Dublin and was in 1827 before the end of his studies professor of astronomy and royal astronomer (Royal Astronomer) for Ireland.

In his early years Hamilton dealt with ray systems and geometric optics . From this, in several publications in 1834 and 1835, he developed the formulation of mechanics that today bears his name (see Hamiltonian mechanics ) . Later he concentrated his investigations on quaternions (hypercomplex numbers), which are used today, for example, in computer graphics .

## life and work

Hamilton was born in Dublin at 36 Dominick Street to the lawyer Archibald Hamilton. His ancestors came from Killyleagh Castle , County Down ; his grandfather was Archibald Hamilton Rowan . Raised by his uncle, Anglican priest and linguist James Hamilton, he soon turned out to be a child prodigy . By the age of five he had knowledge of Latin , Greek and Hebrew and by the age of 13 he had already mastered twelve languages, including the classical and modern European languages ​​as well as Persian , Arabic , Hindi , Sanskrit and Malay . Until the end of his life he often read Persian and Arabic texts for relaxation.

Hamilton's mathematical development seems to have come about completely without the involvement of others, so that his later writings cannot be assigned to any particular school, at best a separate “Hamilton School”. The young Hamilton was not only an excellent mental calculator , but also seemed to find particular fun at times calculating complex formulas down to the last decimal place.

At ten he devoured a Latin edition of Euclid , and at twelve he took up Newton's Arithmetica universalis as an introduction to modern analysis .

At the age of twelve (1817) he challenged Zerah Colburn , a thirteen-year-old boy with "arithmetic genius" who performed in Dublin, against whom he was defeated.

Later he read Clairauts algebra , Newton's Principia and the extensive celestial mechanics of Pierre Simon de Laplace in which he discovered a bug in February 1822 that the attention of the Royal Astronomer of Ireland John Brinkley drew upon him, who predicted a great future as a mathematician .

Hamilton's career at Trinity College Dublin was unprecedented. Among the above-average competitors, he was the first in every subject and in every examination. He was one of the few who achieved top grades in the classical languages ​​Greek and Latin as well as in the natural sciences (an "Optime" in both subjects in the first year of 1823 and 1826). Before graduating, he published his work On Caustics in 1824 , in which he introduced his characteristic function (later named by Heinrich Bruns Eikonal ) in optics , followed by the first part of his pioneering work Theory of Systems of Rays . In 1827 he was appointed professor of astronomy at Trinity College before his final exams, which was linked to the successor of Brinkley, who later became bishop, as the Royal Astronomer of Ireland. His office was the observatory in Dunsink . In 1832 he was elected to the American Academy of Arts and Sciences , in 1864 to the National Academy of Sciences . He had little idea of ​​the practical side of astronomy and was also not interested in this science. On the other hand, he was only expected to use his time as useful as possible in the service of scientific progress, with no stipulations, for example, on practical observation work at the telescope .

On a gentlemanly trip that he undertook to the United Kingdom before taking up his post, he met the poet William Wordsworth , who also paid him a return visit to his observatory, but advised Hamilton, who tried as a hobby poet, to pursue a scientific career ; he found the poems of Hamilton's sister Eliza far more convincing. After his childhood sweetheart Catherine Disney made a financially more advantageous match , Hamilton married Helen Maria Bayley, who came from an estate adjacent to the observatory. However, the marriage, which resulted in three children, was unhappy and the couple lived apart for many years. Personal problems from his marriage and the repeated, frustrating contact with his childhood sweetheart also led to Hamilton's increasing alcohol problems, which were also made public at a banquet in 1845. As a result, he tried to remain celibate for a while, but only succeeded for two years.

In 1834 he transferred his characteristic function as an active function to dynamics and laid new foundations in theoretical mechanics with On a General Method in Dynamics , which later became known as the Hamiltonian theory . In 1835 he was secretary of the British Association for the Advancement of Science and was knighted as a Knight Bachelor . Greater honors quickly followed. In the same year he received the Royal Medal from the Royal Society . In 1837 he was elected President of the Royal Irish Academy and corresponding member of the Academy of Saint Petersburg . In 1839 he was accepted as a corresponding member of the Prussian Academy of Sciences and in 1844 of the Académie des Sciences .

Memorial plaque on the Broom Bridge with the rules of multiplication of the quaternions

After he had already interpreted the complex numbers as ordered pairs of two real numbers in 1833 , he looked for a generalization to three " dimensions ", which is to be understood literally, since he attached a philosophical or geometric dimension to algebra . Influenced by Immanuel Kant , he wrote Algebra, the Science of Pure Time , in 1838 . He was not able to find the extension he was looking for - not to three, but to four dimensions - however, in 1843 when he invented the quaternions on a walk along the Royal Canal on October 16 . Spontaneously he scratched their definition using the multiplication rules of their units 1, i, j, k:

${\ displaystyle \, i ^ {2} = j ^ {2} = k ^ {2} = ijk = -1}$

to Broome Bridge or Brougham Bridge, which was honored in 1958 by the Royal Irish Academy with a plaque on the bridge. In his own words, the idea came to him when, instead of thinking of expanding to three dimensions, he realized that four dimensions were necessary. Hamilton saw a revolution in theoretical physics and mathematics in the quaternions and spent the rest of his life trying to propagate their use, with the assistance of other British mathematicians such as Peter Guthrie Tait in the second half of the 19th century . After his death he left behind an unfinished two-volume work on quaternions, written with the elements of Euclid in the back of his mind. Ultimately, however, vector calculation and vector analysis prevailed as a description language, represented by Hermann Graßmann , Josiah Willard Gibbs and Oliver Heaviside , for example . Lord Kelvin wrote: Hamilton invented quaternions after his really important work was completed. Although beautiful and ingenious in origin, they have been a curse to anyone who has come into contact with them in any way . In his own books, Kelvin avoided both quaternions and vectors. Later it turned out that Olinde Rodrigues also found the quaternions as early as 1840.

Hamilton died in 1865 after a severe attack of gout . He was buried in Mount Jerome Cemetery in Dublin.

The Hamilton crater is named after him.

## literature

• Hamilton Works , 3 vols., 1931-1967
• Hamilton On a general method in dynamics . Dublin 1834
• Hamilton Second Essay On a General Method in Dynamics. In: Philosophical Transactions of the Royal Society of London 125 (1835), pp. 95-144.
• Hamilton Lectures on Quaternions . Dublin 1853
• Hamilton Elements of Quaternions . London 1866. (German: Leipzig 1882–1884, translated by Paul Glan )
• WR Hamilton's treatises on radiation optics , Leipzig, Akademische Verlagsgesellschaft 1933 (edited and translated by Georg Prange )
• Robert Perceval Graves "Life of Sir William Rowan Hamilton", 3 vols., 1882, 1885, 1889, reprint 1975 (with many letters from Hamilton and many of his poems), older biographies can also be found in William Rowan Hamilton: Some Nineteenth Century Perspectives
• Thomas L. Hankins "Sir William Rowan Hamilton," Baltimore: The Johns Hopkins University Press, 1980, 2004, and his article in the Dictionary of Scientific Biography on Hamilton
• S. O'Donnell "William Rowan Hamilton. Portrait of a Prodigy ', Dublin, 1983
• Goldsmith, Dimitric Hamilton , in the series Mathematical Tourist in Mathematical Intelligencer 1989, No. 2 (photo of the bridge where he discovered his quaternions, with the plaque)
• JL Synge, "The life and early work of Sir William Rowan Hamilton," Scripta Mathematica, Vol. 10, 1944, pp. 13-24
• MacDuffee, "Algebra's debt to Hamilton," Scripta Mathematica, Vol. 10, 1944, pp. 25-35.
• JL Synge "Geometrical optics- an introduction to Hamilton's Method", Cambridge University Press 1937
• Bartel Leendert van der Waerden , "Hamilton's discovery of quaternions", Mathematics Magazine, Vol. 49, 1976, pp. 227-234.
• J. Lambek "If Hamilton had prevailed: quaternions in physics", Mathematical Intelligencer, Vol. 17, Issue 4, 1995, pp. 7-15.