Even as Thompson published his revolutionary model for
the atom, Maxwell building on Thompson's earlier explorations of
the underlying properties of
this "aetheric fluid," was well on the way to devising a highly successful
"mechanical" vortex model of the "incompressible aether" itself, in
which Thompson's vortex atom could live -- a model derived in part from the
laboratory-observed elastic and dynamical properties of solids.
Ultimately, in 1873, he would succeed in uniting a couple hundred years of
electrical and magnetic scientific observations into a comprehensive,
overarching electromagnetic theory of light vibrations ... carried
across space by this "incompressible and highly stressed universal aetheric
fluid ..."
Maxwell's mathematical basis for his triumphant unification of these two great mystery forces of 19th Century physics were "quaternions" -- a term invented (adopted would be a more precise description) in the 1840s by mathematician Sir William Rowan Hamilton, for "an ordered pair of complex numbers" (quaternion = four). Complex numbers themselves, according to Hamilton's clarifications of long-mysterious terms such as "imaginary" and "real" numbers utilized in earlier definitions, were nothing more than "pairs of real numbers which are added or multiplied according to certain formal rules." In 1897, A.S. Hathaway formally extended Hamilton's ideas regarding quaternions as "sets of four real numbers" to the idea of four spatial dimensions, in a paper entitled "Quaternions as numbers of four-dimensional space," published in the Bulletin of the American Mathematical Society [4 (1887), 54-7].
It is obvious from Maxwell's own writings that, even before Hathaway's formalization, his choice of quaternions as mathematical operators for his electromagnetic theory was based on his belief that three-dimensional physical phenomena (including even perhaps the basis of human consciousness itself) are dependent upon higher dimensional realities. For, in honor of another great mathematician of the time, multi-dimensional geometer Arthur Cayley, Maxwell wrote ...
"Oh WRETCHED race of men, to space confined!
What honour can ye pay to him, whose mind
To that which lies beyond hath penetrated?
The symbols he hath formed shall sound his praise,
And lead him on through unimagined ways
To conquests new, in worlds not yet created.
First, ye Determinants! In ordered row
And massive column ranged, before him go,
To form a phalanx for his safe protection.
Ye powers of the nth roots of - 1!
Around his head in ceaseless* cycles run,
As unembodied spirits of direction.
And you, ye undevelopable scrolls!
Above the host wave your emblazoned rolls,
Ruled for the record of his bright inventions.
Ye cubic surfaces! By threes and nines
Draw round his camp your seven-and-twenty lines-
The seal of Solomon in three dimensions.
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