- ALGEBRAIC STRUCTURES 453
likely it was his physical intuition that led him to make this discovery. A rotation
in two-dimensional space requires only one parameter for its determination: the
angle of rotation. In three-dimensional space it is necessary to specify the axis
of rotation by a point on the unit sphere, and then the angle of rotation, a total
of three parameters. Dilations then require a fourth parameter. Thus, although
quaternions were invented to describe transformations of three-dimensional space,
they require four parameters to do so.
Hamilton, an excellent physicist and astronomer, worried about simply making
up symbols out of his head and manipulating them. He soon found applications of
them, however; and a school of his followers grew up, dedicated to spreading the
lore of quaternions. By throwing away one of Hamilton's dimensions (the one that
contained the unit) and using only the three symbols i, j, and fc, the American
mathematician Josiah Willard Gibbs (1839-1903) developed the vector calculus
as we know it today, in essentially the same language that is used now. In the
language of vectors quaternions can be explained easily. A quaternion is simply the
formal sum of a number and a point in three-dimensional space, such as A = a + a
or  — b + â. As Hamilton had done with complex numbers, it is possible to
rationalize this seeming absurdity by regarding the number, the point, and the
quaternion itself as quadruples of numbers: á = (á,0,0,0), a = (0,áé,á 2 ,è3),
A = (á,áé,á·2,á 3 ). The familiar cross product developed by Gibbs is obtained by
regarding two vectors as quaternions, multiplying them, then setting the numerical
part equal to zero (projecting from four-dimensional space to three-dimensional
space). Conversely, quaternion multiplication can be denned in terms of the vector
(dot and cross) products: AB = (ab - á- â) + (áâ + ba + á ÷ â). The quaternion
A = a - a is the conjugate, analogous to the complex conjugate, and has the
analogous property AA — o? + a • a = a^2 + \a\^2 — a^2 + a^2 + a^2 + a^2. Thus AA
represents the square of Euclidean distance from A to (0,0,0,0) and can be denoted
\A\^2. This equation in turn shows how to divide quaternions, multiplying by the
reciprocal: l/A = (l/\A\^2 )A. The absolute value of a quaternion has the pleasant
property that \AB\ = \A\ \B.
The Harvard professor Benjamin Peirce (1808-1880) became an enthusiast of
quaternions and was already lecturing on them in 1848, only a few years after their
invention. Like many mathematicians before and after, he was philosophically
attracted to algebra and believed it encapsulated pure thought in a way that was
unique to itself. His treatise Linear Associative Algebra was one of the earliest
treatises in this surprisingly late-arising subject.^17
On the Continent algebra developed from other roots, more geometric in nature,
exemplified by Grassmann's Aiisdehnungslehre, which was described in Section 3 of
Chapter 12. To some Continental mathematicians, what the British were doing did
not seem sufficiently substantial. On New Year's Day 1875, Weierstrass wrote to
his pupil Sof'ya Kovalevskaya that she had much more important things to learn
than Hamilton's quaternions, whose algebraic foundations, he said, were of a very
trivial nature. In his discussion of quaternions Klein [1926, p. 182) remarked,
"It is hardly necessary to mention that the Grassmannians and the quaternionists
were bitter rivals, while each of the two schools in turn split into fiercely warring
subspecies." Weierstrass himself, in 1884, gave a discussion of an algebra, including
(^17) I say "surprisingly" because, as anyone would agree, its basic subject matter linear
equations—is much simpler than many parts of algebra that developed earlier.