Beyond Quaternions (4Nions)
According to Michael J. Crowe [1], James Clerk Maxwell was a brilliant Scot who was very influential in increasing the awareness of the need for a vectorial approach to the solution of physical problems. The most important historical development seems to have been the mathematical classification of physical quantities so that analogies can be applied between scientific analyses of physical phenomena. Maxwell apparently appreciated the utility of serendipity in scientific advances: "Thompson saw that distributions of electrostatic forces in a region containing electrified conductors produced a vector field and that this vector field was analogous to the vector field associated with the flow of heat in an infinite solid." Crowe says, "...the eventual emergence of the vector analysis should be intimately associated with the achievements of the great nine-teenth century theoretical electricians, especially Maxwell. Much is symbolized in the fact that Maxwell first presented his famous equations in the 1860's, written in component notation, whereas in his 1873 TREATISE ON ELECTRICITY AND MAGNETISM he wrote them both in component and in quaternion notation."
Maxwell's views, however, are summed up in Crowe's statement that, "Maxwell favored quternion analysis for the naturalness of its representation of physical entities and for the abbreviations stemming from this. Most of all he favored quaternions because the physical entities were kept before the eye of the mathematician. He was particularly impressed by the operator [2] and the linear vector function. On the other hand, Maxwell in general disliked quaternion "methods" (as opposed to quaternion "ideas"); thus for example he was troubled by the nonhomogeneity of the quaternion or full vector product and by the fact that the square of a vector was negative, which in the case of the velocity vector make kinetic energy negative." In a letter to an academian, Maxwell wrote [3], "I am getting converted to quaternions, and have put some in my book ..." and he even authored a poem To The Chief Musician Upon Nabla in reference to Sir W. R. Hamilton, the discoverer of "4nions" and Professor Tait, the chief proponent.
According to Kelland and Tait [4] in their extensive quaternion text, If α and β are defined as unit vectors, the product of the two vectors Tα and Tβ that are the angle Θ apart is:
αβ=TαTβ(-cosΘ + εsinΘ) = Sαβ + Vαβ Equation 1.
so that:
Sαβ = -TαTβcos Θ Equation 2.
Vαβ = TαTβεsin Θ Equation 3.
where S is read scalar and V is read vector. A coordinate system is defined in the right-hand manner with β along the i axis, γ along the j axis, and ε along the -k axis. The text point out that, "the coefficient of ε in Vαβ is the area of the parallelogram whose sides are equal and parallel to the lines of which α, β are the vectors." The quaternion is generally composed of a real part and a vector. Hamilton is quoted on page 31 of Crowe, "Regarded from a geometrical point of view, this algebraically imaginary part of a quaternion has thus so natural and simple a signification or representation in space, that the difficulty is transferred to the algebraically real part; and we are tempted to ask what this last can denote in geometry, or what in space might have suggested it."
James Arthur Adkins, in Chapter I of his UTA Master's Thesis [5], attributes the development of modern vector analysis to improvements upon quaternion ideas by J. Willard Gibbs and Oliver Heaviside in the late nineteenth century. However, he finds that Herman Grassmann's work DIE AUSDEHNUNGSLEHREcontains brilliant but undeveloped techniques for manipulating vectors that were utilized by William Kingdon Clifford. Adkins concludes in his final chapter that the Clifford Algebra is superior to conventional techniques for ease of manipulation and representation of the electromagnetic field equations.
Maxwell formally expresses his general equations of the electromagnetic field in twenty equations with twenty unknowns in his 1864 paper A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD[6]. The notation and recognition of the importance of the combination of derivitives known today as the curl and divergence are attributed to Maxwell in the equations recorded by Feynman [7]:
с2 ΧΒ = (j/ε0 ) + ∂Е/∂ŧ Equation 7.
The historical development and usage of these important electromagnetic field equations is seen to span the development and discard of quaternion analysis in the favor of the modern vectorial analysis methods. However, the spirit of Maxwell's search for cogent presentation of physical phenomena and scientific advancement through inspired analogies appears to be continuing in the contemporary investigation of the Clifford Algebra.
References
- Crowe, Michael J., 1967, A HISTORY OF VECTOR ANALYSIS, (University of Notre Dame Press, Notre Dame), Chapter Four, UTA Library QA261C779.
- Weisstein, Eric W. "Curl." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Curl.html
- Campbell, Lewis and Garnett, William, 1969, THE LIFE OF JAMES CLERK MAXWELL, (Johnson Reprint Corporation, New strategies York), p.383, p.634, UTA Library QC16M4C21969.
- Kelland, M. A. and Tait, P. G., 1882, INTRODUCTION TO QUATERNIONS, (Macmillan And Co., London), Chapter 3, UTA Library QA196K41882.
- Adkins, James Arthur, 1977, APPLICATIONS OF CLIFFORD ALGEBRAS TO PHYSICAL PROBLEMS INVOLVING ROTATION OF COORDINATES, (The University Of Texas At Arlington, Arlington), Chapters I, IV, UTA Library LD5315 1977A33C2.
- Niven, W. D., 1965, THE SCIENTIFIC PAPERS OF JAMES CLERK MAXWELL, VOLUME ONE, (Dover Publications, New strategies York), pp.526-597, UTA Library QC3M47 1965.
- Feynman, Richard P., THE FEYNMAN LECTURES OF PHYSICS, VOLUME II, 1964, (Addison-Wesley Publishing, Dallas), Chapter 18, UTA Library QC23F47V2C7.
