Editing Geometric algebra (section)

=== Before the 20th century ===
Although the connection of geometry with algebra dates as far back at least to [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' in the third century B.C. (see [[History of elementary algebra#Greek geometric algebra|Greek geometric algebra]]), GA in the sense used in this article was not developed until 1844, when it was used in a ''systematic way'' to describe the geometrical properties and ''transformations'' of a space. In that year, [[Hermann Grassmann]] introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the [[propositional calculus]]) that encoded all of the geometrical information of a space.{{sfn|ps=|Grassmann|1844}} Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them being [[Euclidean space]], [[affine space]], and [[projective space]]. Following Grassmann, in 1878 [[William Kingdon Clifford]] examined Grassmann's algebraic system alongside the [[quaternions]] of [[William Rowan Hamilton]] in {{Harvard citation|Clifford|1878}}. From his point of view, the quaternions described certain ''transformations'' (which he called ''rotors''), whereas Grassmann's algebra described certain ''properties'' (or ''Strecken'' such as length, area, and volume). His contribution was to define a new product – the ''geometric product'' – on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently, [[Rudolf Lipschitz]] in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in {{tmath|1= n }} dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.

Nevertheless, another revolutionary development of the 19th-century would completely overshadow the geometric algebras: that of [[vector analysis]], developed independently by [[Josiah Willard Gibbs]] and [[Oliver Heaviside]]. Vector analysis was motivated by [[James Clerk Maxwell]]'s studies of [[electromagnetism]], and specifically the need to express and manipulate conveniently certain [[differential equation]]s. Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adopted it as their geometrical toolkit of choice, particularly following the influential 1901 textbook ''[[Vector Analysis]]'' by [[Edwin Bidwell Wilson]], following lectures of Gibbs.

In more detail, there have been three approaches to geometric algebra: [[quaternion]]ic analysis, initiated by Hamilton in 1843 and geometrized as rotors by Clifford in 1878; geometric algebra, initiated by Grassmann in 1844; and vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis can be seen in the use of {{tmath|1= i }}, {{tmath|1= j }}, {{tmath|1= k }} to indicate the basis vectors of {{tmath|1= \mathbf{R}^3 }}: it is being thought of as the purely imaginary quaternions. From the perspective of geometric algebra, the even subalgebra of the Space Time Algebra is isomorphic to the GA of 3D Euclidean space and quaternions are isomorphic to the even subalgebra of the GA of 3D Euclidean space, which unifies the three approaches.