I have to apologize for the fact that I haven't posted any new Pynchon material recently. I still am working on the essay/science primer for Against the Day, however the task has turned out to be more formidable than I first anticipated. The problem is this: there is so much material that could be included in a primer on vectors, quaternions, space-time, etc., much more than I have the time or, in some cases, expertise to cover. In order to keep things focused on the truly relevant science, I must finish a second reading of the book.
I'm well on my way through that second reading, furiously taking notes, and I'm working on various portions of the essay, but it will probably be at least a month before I finish and can post my results.
In the meantime, I highly recommend this book, which I'd bet was one of Pynchon's sources on quaterions and vector analysis (as you can imagine, there just aren't that many books out there on the history of vector analysis): A History of Vector Analysis. It looks like it's hard to find - I picked up a copy from my local university library. I'll leave you with a few choice excerpts from the book that relate to what Hamilton was trying to accomplish with quaternions, and how his efforts were viewed:
p. 37, from an 1857 review of Hamilton's work on quaternions:
"It is confidently predicted, by those best qualified to judge, that in the coming centuries Hamilton's Quaternions will stand out as the great discovery of our nineteenth century."
[In reality, quaternions were eclipsed by vector analysis only a few decades later.]
p. 23-24: From an essay published by Hamilton in 1837, a section called "On Algebra as the Science of Pure Time" (to understand this passage, recall that imaginary numbers are multiples of the square root of -1):
"The thing aimed at, is to improve the Science, not the Art nor the Language of Algebra. The imperfections sought to be removed, are confusions of thought, and obscurities or errors of reasoning; not difficulties of application of an instrument nor failures of symmetry in expression...
"For it has not fared with the principles of Algebra as with the principles of Geometry. No candid and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by EUCLID in his Elements, two thousand years ago... The doctrine involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the plan of argument.
"But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers each denoting magnitudes less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive or negative, nor yet null numers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing. It must be hard to found a SCIENCE on grounds such as these..."
[Note in this upcoming section that Hamilton speculates that with geometry as the science of space, perhaps algebra could become the science of time:]
Hamilton asks "whether existing Algebra, in the state to which it has been already unfolded by the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of developing a SCIENCE of Algebra: a Science properly so called; strict, pure, and independent; deduced by valid reasonings from its own intuitive principles; and thus not less an object of priori contemplation than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and from the Signs by which it may express its meaning..."
He suggests that "the Intuition of TIME is such a rudiment... The argument for the conclusion that the notion of time may be unfolded into and independent Pure Science, or that a Science of Pure Time is possible, rests chiefly on the existence of certain priori intuitions, connected with that notion of time, and fitted to become the sources of a pure Science; and on the actual deduction of such a Science from those principles, which the author conceives that he has begun."
Hamilton here draws a comparison between Euclidean geometry as a pure science of space, and his efforts to make algebra a pure science of time. Historically, and in light of Against the Day, it is interesting to note that Hamilton wrote this before Non-Euclidean geometry became widely known (after 1860, according to Crowe), and long before experiments suggested there was anything wrong with our intuitive notions of time which Hamilton wanted to rely on. In essence, Hamilton's quaternions and Euclidean geometry are part of a classical world that came to an end during the time frame of Pynchon's book. (In the long run, vector analysis and quaternions themselves, instead of being a science of time, became an algebra dealing with space.)