As some know, my primary research work is in the area of nodal aberration theory. The entry point to this aberration theory of optical systems with typically spherical or conic surfaces but without positional rotational symmetry is necessarily based in a mathematical operation introduced by R.V. Shack, my PhD advisor from 1978-1980, which he and I called vector multiplication. At one point, I recall a rare outing, on foot, with Prof. Shack on the campus of the University of Arizona, in search of a mathematician who could point to a reference for this operation.
The operation can be accomplished by invoking the mathematics of complex numbers, but aberration theory in general has nothing to do with complex numbers. When there is no symmetry however, it does have something to do with vectors, typically an aperture vector in the pupil of the optical system and a field vector in the image (or object) plane. In most applications of interest, the plane containing the pupil and the plane containing the image are parallel, but not coincident. The operation in question, detailed in my 2005 JOSA A article, is to “multiply” either two identical aperture vectors, or two identical field vectors, or, in the more interesting case, a coplanar aperture and field vector. The result of this operation, which is not described in any reference on vector algebra (that I have found so far), is another vector, coplanar with the two vectors forming the operation, which do, by definition, always have a common origin, at least in projection. In that it is a vector, it is not a dot-product, and, in that it continues to be coplanar with the two constructing vectors, it is not a cross-product. The resultant vector has the same common origin, has an orientation angle that is the sum of the individual orientation angles of the two vectors being multiplied (notably from an axis which is by definition the zero orientation angle for the vector), and the magnitude is the product of the magnitudes. In the course of writing this, I have been directed to Visual Complex Analysis, by Needham, as perhaps an acceptable merging of complex numbers with planar vector analysis. I will look into this, perhaps even next.
It continues to fascinate me that there appears no other entry point to nodal aberration theory, which is clearly (to me at least) the most fundamental and general form of a description of optical aberrations in imaging systems, than to invoke an operation that essentially does not exist in mathematics as it is currently taught. Actually, there is no mathematics for a physics characterized by functions with multiple distributed zeros, at this time (that I’m aware of). Here though I expect it is ignorance on my part. For those following this blog from the beginning, I wrote of my personal discovery of the work of Hestenes and Geometrical/Clifford algebra last year. At the time, I thought that that work contained, somewhere, the operation Prof. Shack invoked. However, I return to the topic today, to report that after reading A History of Vector Analysis, by Michael J. Crowe (University of Notre Dame Press, 1967), I’m not so sure anymore.
Crowe’s book is fascinating to me in more than one way. I have said earlier that the field of optical design came to be around 1880 when Zeiss, Abbe, and Schott worked together to first engineer optical glass and then to use it to create new instrumentation, starting with the microscope. So while various forms of aberration theory appear earlier (I am still working on this timeline), starting it appears with Airy in the early 1800s and including the likes of Gauss, Coddington, Petzval, Seidel, and perhaps culminating (at least according to Andrew Rakich, and I tend to agree) with Schwarzschild in 1905, the industry came along somewhat later. What really surprises me though is that vector mathematics, in fact, did not even exist in the modern form in 1900. Until now, I viewed mathematics, particularly vector mathematics, as existing from long, long ago. Crowe covers the period from about 1840 to about 1910, and focuses on in particular the work of Hamilton (as in the optics term Hamiltonian) and a lesser-known German, Grassmann. Hamilton spent over 20 years and a majority of his energies attempting to create a mathematics that was termed quaternions (his book is available as a free download from Google Scholar and according to any and all scientists quoted in Crowe is literally impossible to read all the way through).

Cover page of one of Hamilton’s two major works on Quarterions
Although I now know their history, I would be hard pressed to explain what “they” are. My simplistic takeaway is this is a vector-like mathematics where the local coordinate system can be invoked on-the-fly and can be changed even within an equation. Therefore, the equations contain embedded notation that conveys the local coordinate reference for each mathematical operation. Although I have not found a book of the likes of Crowe, it appears that the period from 1910, where the scalar dot product and vector cross product and the divergence function does not yet exist in the modern form, to 1930 is the defining period for the acceptance of the modern form of vector operations and, to some extent, the death of quaternions (as a path within planar vector analysis), until recently. At some point, someone named Clifford (perhaps a Clifford described by Crowe as a short-lived scientist who died young in 1879) comes along and establishes a basis for a now very obscure branch of quaternions that survives today with the work of people like Hestenes, whom I mentioned in an earlier piece. However, at the point of this writing, it appears that what has come to be called Geometric Algebra, or Clifford Algebra, does not in fact contain, define, use, or describe the operation that Prof Shack invoked to discover the fundamental form of optical aberrations.
The closest I found in Crowe is actually the first reference, on page 7, to the work of an obscure surveyor/mathematician, Wessel. Somewhat interesting is that this is also, according to Crowe, the first reference to vectors (as in oriented lines), in 1799 and is based in the mathematics of complex numbers and the mathematics of, for the first time, inclined lines. Previously, the only considerations in print described only parallel and anti-parallel lines. As least based on the words, “…he proceeded to introduce the multiplication of lines. The product of two lines (coplanar with each other and with the positive unit) was to have a length equal to the product of the length of the two factors. The product line was to be coplanar with the two factor lines and was to have its inclination or direction angle (defined by reference to the inclination of the positive unit as zero degrees) equal to the sum of the inclinations of the factor lines“. He in fact described the operation that we have adopted, whereby a vector is multiplied with another to create a third vector that remains in the same plane and not, as is the case with the cross product, in an orthogonal plane. What is most interesting is that this is the only description of this vector multiplication definition that I have found and it does not appear again in Crowe’s book. Most significantly, the mathematics that the remainder of Crowe’s book covers is conceptually based in managing and understanding planar areas bounded by vectors. Since aberration theory has no relevance to the area concept, the mathematics of quaternions or its derivatives are not a reference point.
There was one more historically striking paragraph in Crowe. He points out that Einstein, in the book with Infeld, considered the introduction the concept of a “field” into physics as the most significant event in 19th-century science. This comes up because one of the first people to use the divergence function was Maxwell, in developing his electromagnetic equations (prior to the concept of a field). This is one of the major topics areas in Crowe. So, whereas we are currently a generation who spends much of our time in Flatland, as described by Gamow – 2D thinking – it appears that prior to about 1910, the thinking was purely in the context of lines and points (or the concept of needle space as recently coined by Prof. Jim Leger of U. of MN), which is exactly the scope of quaternion mathematics.
I’m now looking to learn similar information about the history of Clifford, and Clifford Algebra (I hear Geometric Algebra by Lee Dorst is a good choice) and the correlation of visual complex analysis with complex numbers (which sounds promising at the moment). Separately, I hear, and it makes sense, that quaternions have finally been rediscovered and found a solid application – facet management in computer generated AI. Stay tuned – and send feedback.