452 15. MODERN ALGEBRAin the Journal fur die reine und angewandte Mathernatik in 1914. In his introduc-
tion Fraenkei cited the large number of particular examples as a reason for defining
the abstract object. He required that the ring have at least one right identity, an
element å such that áå = a for all a, and that for at least one of the right identities
every regular element should have an inverse. The English term was introduced by
Eric Temple Bell (pseudonym of John Taine, 1883 1960) in a paper in the Bulletin
of the American Mathematical Society in 1930.
Although mathematical communication was very extensive from the nineteenth
century on, there was still enough difficulty due to language and transportation that
British and Continental mathematicians sometimes took very different approaches
to the same subject. Such was the case in algebra, where the solution of equations
and abstract number theory led Continental mathematicians in one direction, at
the time when British mathematicians were pursuing an abstract approach to al-
gebra having connections with an outstanding British school of symbolic logic. For
linguistic and academic reasons, the British approach also caught on in the United
States, the first American foray into mathematical research. This Anglo-American
algebra has been studied by Parshall (1985).
One of the first examples of this British algebra was the algebra of quaternions,
invented by William Rowan Hamilton in 1843. Hamilton had been intrigued by
the complex numbers since his teenage years. He questioned the meaningfulness
of writing, for example 3 + since this notation made it appear that two
objects of different kinds real and imaginary numbers were being added. To
rationalize this process, he took the step that seems obvious now, regarding the
two numbers as ordered pairs, so that 3 is merely an abbreviation for (3,0) and
an abbreviation for (0, \/E), thereby algebraizing the plane. Hamilton was
very much a physicist, and he saw complex multiplication, when the numbers were
put in polar form r(cosf? 4- isinf?), as representing rotations and dilations. For
him, complex addition represented all the possible translations of the plane, and
complex multiplication sufficed to describe all its rotations and dilations.
Influenced by the mysticism of the poet Coleridge (1772-1834),^15 whom he
knew personally, he felt that great insight would be obtained if he could similarly
algebraize three-dimensional space, that is, find a way to multiply triples of numbers
(x, y, z) similar to the complex multiplication of pairs (x, y). In particular, he wished
to find algebraic operations corresponding to all translations and rotations of three-
dimensional space. Translations were not a problem, since ordinary addition took
care of them. After much reflection, during a walk in Dublin on October 10, 1843
that has become one of the most famous events in the history of mathematics, he
realized that he needed a fourth quantity, since if he used one coordinate to provide
a unit x, having the property that xy = y and xz — z, the product yz would have
to be expressible symmetrically in x, y, and z. When the formulas we now write
as i^2 = j^2 = k^2 = -1, ij = —ji = k, ki = —ik = j, jk = —kj = i occurred to him
during this walk, he scratched them in the stone on Brougham Bridge.^16 In his
1845 paper in the Quarterly Journal he referred to ix + jy + kz as "the vector from
0 to the point x,y,z." The word vector (Latin for carrier) occurs in this context
for the first time. A quaternion thus consists of a number and a vector. Very
(^15) Coleridge's most famous poem, The Rime of the Ancient Mariner, is full of mystical uses of
the numbers 3, 7, and 9.
(^16) In the 160 years since then, they have been effaced.