acknowledged today by a plaque. With the date scored into his mind, the subject
became Hamilton’s obsession. He lectured on it year after year and published two
heavyweight books on his ‘westward floating, mystic dream of four’.
One peculiarity of quarterions is that when they are multiplied together, the
order in which this is done is vitally important, contrary to the rules of ordinary
arithmetic. In 1844 the German linguist and mathematician Hermann Grassmann
published another algebraic system with rather less drama. Ignored at the time,
it has turned out to be far reaching. Today both quaternions and Grassmann’s
algebra have applications in geometry, physics and computer graphics.
The abstract
In the 20th century the dominant paradigm of algebra was the axiomatic
method. This had been used as a basis for geometry by Euclid but it wasn’t
applied to algebra until comparatively recently.
Emmy Noether was the champion of the abstract method. In this modern
algebra, the pervading idea is the study of structure where individual examples
are subservient to the general abstract notion. If individual examples have the
same structure but perhaps different notation they are called isomorphic.
The most fundamental algebraic structure is a group and this is defined by a
list of axioms (see page 155). There are structures with fewer axioms (such as
groupoids, semi-groups and quasi-groups) and structures with more axioms (like
rings, skew-fields, integral domains and fields). All these new words were
imported into mathematics in the early 20th century as algebra transformed itself
into an abstract science known as ‘modern algebra’.