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e-mail: math@ H Y P E R C O M P L E X .COM H << | >> T.Nabla 1846 THE NABLA OF SIR WILLIAM ROWAN

HAMILTON ( revision .02, 2001-07-19, new section 2002-05-07 )

Sir (1805-1865), Irish Mathematician and Physicist, discovered , q = w + ix + jy + kz, in 18431, and in 18462 he introduced the differential operator, = i@/@x + j@/@y + k@/@z, to facilitate the special vector differentiation of quaternions. His original symbol, the "horizontal wedge," , was later turned over after his death by mathematicians3 and written as a "vertical wedge," , instead. The came into widspread use, while the names4 "nabla" and "del" were assigned to this operator, and beginning in 1881 the definition of the vector (i,j,k) was also changed, this by physicists5 who objected to Hamilton's mathematical definition of vectors and sought what they thought was a more physically meaningful definition, so that today, the we know, and are so familiar with, bears little resemblence to Hamilton's original differential operator.

While Hamilton's Operator may be somewhat unfamiliar to us, we can easily grasp the essential difference between this original operator and its modern incarnation as found in our modern . If we agree to use the simple trick6 of matching up only the components of the old and new vectors, then for a vector A = iA1 + jA2 + kA3, we can write;

A = -div(A) + (A)

A = -div(A) - curl(A)

where "div()" and "curl()" are the usual differential operations in modern . Care must be taken when using these expressions on either side of the equal signs as equivalents, but with the appropriate warnings6 heeded they can be very useful in facilitating easy recognition of results in terms of things we already know from modern vectors. This allows us to quickly grasp the results of quaternion analysis, something that could otherwize be a rather tedious affair. A glance is all we need to understand the significance of the result. Hamilton's operator produces the sum of a and vector. The vector part changes when the direction of the operation is reversed, because quaternions do not commute, but the scalar part is unchanged in sign. This means one cannot indicate the difference of differentiation from left and right with a simple overall minus sign like in cross products7.

Mysteriously, although Hamilton dies as much as nineteen years after he first introduces this operator, during which time he is mostly devoted to the development of quaternions, and spends all of his time in the last years working on the Elements8, almost nothing can be found in Hamilton's writings on the topic, apart from two brief references to this operator. Charles Jasper Joly writes9 - "Hamilton's writings on the operator consist, so far as I am aware, and I have searched through his manuscripts in the library of Trinity College, of a communication to the (July 20, 1846), which is published in the proceedings, Vol.iii, p.291, and practically reprinted in the Phil.Mag. of the following year, and of Art.620 of the Lectures on Quaternions."10

Apparently, Hamilton dies before he gets the opportunity to write down what his thoughts are on this differential operator. As Joly says in his preface to Hamilton's Elements11 - "Sir William Rowan Hamilton died on the 2nd of September, 1865, leaving his great on Quaternions unfinished. He intended to have added some account of the operator , an Index, and an Appendix containing notes on Anharmonic Coordinates, on the Barycentric Calculus, and on proofs of his geometrical theorems stated in Nichol's Cyclopaedia. At the time of his death, with the exception of a fragment of the preface, and a small portion of the table of contents, all the manuscript he had prepared was in type. As he rarely commenced writing before his thoughts were fully matured, he has left no outline

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of the additions contemplated."

Not only could Joly not find any further reference to , but even in refering to this operator he is already using the modern vertical wedge symbol, , a symbol Hamilton chose not to use, as if they were totally interchangeable. And although he does remark later in footnotes of the 2nd edition of the Elements that Hamilton's symbol was different, and renders the symbol correctly then as , Joly nevertheless adopts the modern mathematicians vertical wedge in all his own writings, consistent with everyone else of the , and never seems to be aware of the subtle difference in meaning introduced by the new symbol, even though he is forced to invent an alternate mechanism to distinguish left and right differentiations later in Art.57 of his own 1905 work A Manual of Quaternions12.

This modern vertical wedge was in fact considered by Hamilton himself, and actually used for the differential operator in Hamilton's very first communication on the subject. The paper, prepared for a meeting of the British Association at Southampton, never made it to its intended destination, and so was never published. But, we know of it because Hamilton refers to this first paper when discussing the operator in the October 1847 edition of the . On page 291, Hamilton writes13, "In the paper* designed for Southampton it was remarked, as an illustration, that this result enables us to put the known thermological equation,

d2v/dx2 + d2v/dy2 + d2v/dz2 + a dv/dt = 0, under the new and more symbolic form,

( 2. - a d/dt)v = 0; . . . . . (d.) when v denotes, in quantity and in direction, the flux of heat, at the time t and at the xyz."

And on this same page, in the corresponding footnote to the referenced Southampton 'paper*', he explains, "* In that paper itself, the characteristic was written ; but this more common sign has been so often used with other meanings, that it seems desirable to abstain from appropriating it to the new signification here proposed." Hamilton feels the need to explain here why he has just changed the symbol from , which he had only very recently used in that Southampton Paper, and is now writing instead, giving a plausible argument. But, his explanation is a bit puzzling, since no one else is using the symbol for anything. Where are all these other meanings that cause confusion? Only Hamilton has used this symbol before, and he used it in two different contexts, once as a permutation symbol, and once as an arbitrary function, but certainly "so often used with other meanings" is definitely an exaggeration on his part here. Why did he feel the need to exaggerate? He was constructing a plausable argument for his readers, because he was not prepared to reveal his ideas and the real reason for making the change, as we shall argue below.9

The Southampton paper was never published. So in fact, all Hamilton's public writings use , and were it not for that one footnote, we would not have known that Hamilton actually considered and rejected it in favor of the rotated . We get an explanation of what happened to the Southampton Paper from Hamilton's very extensive biography written by Graves; it was misdirected to Sir John Herschel's home at Collingwood and never made it to Southampton.

In discussing the year 1846, Graves writes14, "From the letters which passed at this time between Hamilton and Herschel, I gather that Hamilton was prevented by the illness, which proved the mortal illness, of his sister Eliza from attending the Meeting of the Association at Southampton, and that he sent to Herschel, as President of Section A, a communication 'respecting the application of my new algebraic geometry to the mathematics of Heat, and to some parts of the general theory of and surfaces, especially as respects Ellipsoids and Cones of the Second Degree.' The communication, through mistake, was not forwarded from Collingwood; and Herschel, finding it there on his return home, writes to express his regret, and adds, 'it looks very beautiful. Go on and prosper.'"

Peter Guthrie Tait (1831-1901), the Scottish Mathematician and Physicist, close friend of , and understudy of Hamilton, was the first person to take up Hamilton's Operator and attempt to develop the theory. But, Tait seems not to have understood why Hamilton used the

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symbol, for he was one of the first to change this symbol to the modern , if not the very first person to do so, and it seems he never considered the profound implications of the switch.

In his 1898 Scientific Papers15 Tait writes - "I was originally attracted to the study of Quaternions by Sir W. R. Hamilton's

= i d/dx + j d/dy + k d/dz to which he called special attention (Lectures on Quaternions, $ 620) on account of its promise of usefulness in physical applications. But I soon found that in that its full may be applied, in general investigations, it is necessary that we should have processes of definite integration, of the kinds required in physics, applicable to quaternion symbols and not merely to scalar variables. I often consulted Hamilton about this want, and he promised to endeavour to supply it at some future time. I fancy that shortly before his death he must in some way have supplied it, though he certainly did not print, nor does he appear even to have written, anything on the subject. In one of the last letters I received from him, he said that he intended to conclude the final chapter of his Elements, which is devoted to physical applications, by some sections on the use of the operator mentioned above. That chapter remains unfinished, and as Hamilton rarely wrote down the steps of even a complex train of mathematical reasoning until he had mentally completed it, it is to be feared that this portion of his investigations is entirely lost."16

In refering to Hamilton's Operator, Tait uses the modern here. And throughout the Scientific Papers this vertical wedge is used exclusively. In fact, even when he "reprints" his early papers, he removes all instances of the original symbol and replaces them with , so that his reprints are not exact copies of the originals. One must view the actual Edinburgh Proceedings themselves, for example, years 1862-72, to observe the transformation in this symbol that takes place over time. The last article to use the in the Proceedings was his paper "On the most General of an Incompressible Perfect Fluid," read Monday 21st, March 1870.17

W. R. Hamilton P. G. Tait

A A

A A ?

AB AB

AB AB ?

Polarity18. When the mathematicians turned over the "horizontal wedge," , into the now modern "vertical wedge," , the first thing that was lost was the subtle explicit reference to polarity. Hamilton was a strong believer in the notion that quaternions could accurately capture the inherent polarity found almost everywhere in nature, as exemplified by the left-hand verses the right-hand, hot verses cold temperatures, positive verses negative electric charges, and the north verses south poles of magnets. This binary polarity was part of the philosopher's metaphysical ideas of the fundamental construction of the world, and Hamilton was also a keen student of metaphysics. The first area that Hamilton thought this quaternion polarity would find obvious application was in electric theory. In the 1866 1st Edition of the Elements of Quaternions, his son William Edwin Hamilton writes19, - "Shortly before my father's death, I had several conversations with him on the subject of the 'Elements.' In these he spoke of anticipated applications of Quaternions to Electricity, and to all questions in which the idea of Polarity is involved - applications which he never in his own lifetime expected to be able fully to develop..."

At the June 1845 meeting of the British Association that takes place at Cambridge, Hamilton makes a special note of an observation of his concerning this application of quaternion polarity. Graves writes20, "In connexion with the exposition of his Quaternions which he was called upon to give, Hamilton is stated in the Report of the British Association for this year to have said 'that he wished to have placed on the records the following conjecture as to a future application of Quaternions:-"Is there not an analogy between the fundamental pair of equations ij=k ji=-k, and the facts of opposite currents of electricity corresponding to opposite ?"'"

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The polarity explicit in Hamilton's Operator, comes from the non-symmetrical shape of the horizontal wedge symbol when turning about the vertical, which allows one to readily distinguish the two products, A and A , from each other. And this is necessary to signify the idea that the result of differentiating from the left is different from that when differentiating from the right. Owing to the non-commuting nature of the operation in quaternion , this distinction becomes important. Geometrically, the two different forms utilize two different rotary movements in the differential. In one case a right-hand is involved, and in the other a left-hand rotation occurs instead.

While it is sometimes possible to use the vertical wedge, A and A , to indicate these two products, as Charles Jasper Joly actually does briefly in Art.57 of his 1905 text21 A Manual of Quaternions, it becomes almost impossible to distinguish what is being differentiated when three or more symbols appear in a . How are we to tell what is being differentiated in, AB, for example? Joly gets around this problem by using Greek letters for those parameters not being differentiated, and thus considered quaternion constants, and Roman letters for those that are being differentiated, thus being considered quaternion variables. And for the rather simple expressions he deals with briefly in his book this technique works. But his solution is obviously inferior to the more explicit expressions we can construct using Hamilton's own operator. The two forms, AB and AB, pose no special problems.

In the product, AB, the variable being differentiated, B, is near the "fat end" of the wedge. While the variable, A, that is simply being multiplied by, i.e. the multiplicand, is near the "thin end" of the wedge. It is almost as if the wedge itself were deliberately being reduced in size towards the multiplicand to become the usual dot '.' operator symbol that appears between an ordinary product of two variables like, A.B, and so it is always clear and unambiguous which variable is being differentiated and which multiplied. And therefore when we write the alternate product, AB, we know that it is A that is being differentiated this time, while B is being multiplied. One then thinks of the horizontal wedge as being the of two symbols, a 'dot' and a 'bar', that is, a '.' and a '|', indicating the two operations 'multiplication' and 'differentiation', yoked together into a single symbol that then represents ordinary quaternion multiplication on one side, but more complicated quaternion differentiation on the other side.

The idea of differentiation involves the separation of two points. It refers to the change that occurs from here to there. It utilizes the concept of a in the parameter . The 'bar', or '|', is thus an appropriate symbol for this type of action, as it is very suggestive of that same displacement idea underlying differentiation. While the idea of multiplication, on the other hand, already has traditional 'dot', or '.', as its proper symbol. Thus, Hamilton's horizontal wedge shows much careful thought in its selection.22

James Clerk Maxwell (1831-1879), Scottish Physicist, writes the famous on Electricity and Magnetism23 which is first published in 1873. In a letter24 dated 11 December 1867, Maxwell writes asking Tait about the special form of the symbol for Hamilton's Operator. Maxwell notes that a few other mathematicians have introduced their own symbol variations and alternate notation for very similar mathematical concepts, and he was puzzled as to why the or were chosen for quaternions, rather than, for example, the wedge used by the excellent French mathematician Gabriel Lame. At the very end of this letter Maxwell concludes with a question direct to the issue, "Is there any virtue in turning around 30o ?"

Tait responds to this question in a letter25 to Maxwell dated 13 December 1867, that the "... is required in 4ions for its finite diffce meaning--so we do or for the flux..." The term, 4ions, was Hamilton's close friend John Graves' way of refering to quaternions, and from this response it is clear that Tait had little idea why Hamilton really chose the horizontal wedge. Tait's explanation, that the horizontal and vertical wedge symbols were chosen simply to distinguish Hamilton's Operator from the already accepted finite difference smybol , implying that either one of those other symbols could serve this task, shows his lack of understanding. As just mentioned above, Joly tells us that Hamilton rarely committed anything "to writing before his thoughts were fully matured" ! Even Tait, in his Scientific Papers, as noted above, refers to this characteristic of Hamilton. We can be sure that Hamilton thought very carefully about the choice of symbol, reflected on the characteristics of the obvious alternatives, and picked the best symbol for the task he envisioned for this operator, before

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writing it up and presenting the new idea to the Royal Irish Academy, and including it again seven years later in the 1853 Lectures.26

If Hamilton was only seeking a symbol that could be distinguished from , then was the obvious best choice. Because, the two symbols, and , are sufficiently close to each other that confusion could arise, especially when these symbols are handwritten quickly, as in an exam, or taking notes in class, or even making notes to oneself on this or that idea upon occasion. It is not possible to confuse the vertical wedge with the mathematician's difference symbol. So why didn't Hamilton pick it instead? It seems, Tait's misunderstanding may have contributed to his own switching over to the vertical wedge. As Maxwell points out in his letter, Tait uses Hamilton's original horizontal wedge in his early 1862 writings, but also seems to have used the vertical wedge in other later writings.27 By the time of his publication of An Elementary Treatise on Quaternions in 186728, at least certainly by its 2nd edition in 187529, Tait is writing this operator exclusively as the vertical wedge, and the horizontal wedge dissappears from the mathematical and scientific literature almost entirely, except in occasional footnotes and comments by various authors concerning historical references to the relevant past writings of Hamilton or Tait. But even in reprints of past papers we find this symbol removed and replaced by , much like one would replace a typesetting font for an ordinary character of the alphabet when electing to reproduce an old text. So insubstantial was the significance of Hamilton's original symbol to him that he Tait himself, on editing his own past papers for reprinting in his 1898 Scientific Papers, thought nothing of removing all the original instances of from his own work.

The problem was, of course, Hamilton never actually uses this symbol in context; and does not suggest the left and right applications as we have done above. Apart from simply giving the single definition, = i d/dx + j d/dy + k d/dz, declaring this operator useful, and showing the related forms for the of this operator, he doesn't give any examples of how it is to be used. In Art.620 of the Lectures he writes30; "The bare inspection of these forms may suffice to convince any person who is acquainted, even slightly (and I do not pretend to be well acquainted), with the modern researches in analytical physics, respecting attraction, heat, electricity, magnetism, etc., that the equations of the present article must yet become (as above hinted) extensively useful in the mathematical study of nature, when the calculus of quaternions shall come to attract a more general attention than that which it has hitherto received, and shall be wielded, as an instrument of research by abler hands than mine."

Hamilton's "bare inspection" tells it all. From just looking at the "form" one was expected to be able to tell how to use the operator and apply the operator to the relevant equations found in various fields of physics. This philosophy could easily be seen to apply both to the "form of the equations" to which the coment directly refers, and indirectly to the "form of the operator symbol" used in those equations which he was anticipating to use later in developing quaternion theory. Hamilton's position was that it should be obvious how to interpret these things at a glance. But he neglected to go into the details, thus leaving it up to the reader to actually do the interpreting. And this is where Tait failed.

Maxwell sought out Tait's help in trying to learn more about quaternions, because Tait was as close as anyone could get to Hamilton without actually contacting Hamilton himself, at least on the subject of Quaternions. And by the time Maxwell develops his keen interest in quaternions, around 1870, Hamilton has already long since passed on to the hereafter. But while Tait was close to both Maxwell and Hamilton, he was more of a student of Hamilton's, while he was literally classmates with Maxwell. Much of Tait's ideas on Quaternions came, not from Hamilton, but from Tait himself, reflecting on things Hamilton had said or written, and Tait's own exploration into the . Tait was as much a developer of quaternion theory as Hamilton, and not simply a promoter of this new mathematical art, as other quaternion advocates were. But, Tait did not have the same academic interests as Hamilton, and therefore not the same influences, and so could not share the vision Hamilton had concerning certain things. Hamilton's interests in metaphysics gave him an appreciation for the extensive reverberation simple themes and elementary principles expressed throughout nature. And when he came to construct ideas in mathematics, he sought to once again echo the universal principles in his conceptual and symbolic constructions. The depth of thought behind the constructive ideas coming from Hamilton were simply beyond the limits of Tait's intellectual exposure. So, that which by "bare inspection" was obvious to Hamilton, was simply not at all enlightening to Tait.

Thus, once Tait began to use Hamilton's , he would have found, as a practical matter, that when handwriting notes on the subject, the was certainly more convenient to write to avoid that confusion with . And since the depth of his own thinking could lead him no further than to infer no greater relevance than that the ability to distinguish this operator from that of the finite difference, as

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being the cause in Hamilton's mind, he naturally inclined eventually to the vertical wedge. As an inspection of the Royal Society of Edinburgh Proceedings shows, by the end of the year 1870, this conversion of the symbol appears to have been completed.

Tait's inability to comprehend Hamilton came from his own prejudice against metaphysics, a favorite subject of Hamilton's musings, and the very subject that was likely therefore to be the key to unlocking the meaning of the symbol. Hamilton was not inclined to reveal all his thoughts on the subject of this differential operator, because understandably, they were still fermenting, and not yet in finished form. To try to understand the symbol, one had to examine the conceptual framework Hamilton was working from, and attempt to reconstruct the vision of the world as Hamilton might have seen it. Without metaphysics, how could Tait do this? In his 1877 book31 "Sketch of Thermodynamics," Tait writes, "No a priori reasoning can conduct us demonstratively to a single physical truth." And he goes on to reveal his contempt for metaphysical arguments. He rejects the old metaphysical conception of heat being a form of matter or , declaring it can only be a form of , the motion of matter, and uses this example to declare, "So much for the trustworthness of the metaphysical treatment of a physical question! Such a lesson should never be lost sight of; so deserved and so complete a refutation of the sophistical nonsense of the schoolmen, and so valuable a warning to the 'Philosopher' who may be disposed to a priori argument as more dignified and less laborious than experiment, can scarcely occur again. Even the despised perpetual-motionist has more reason on his side than the metaphysical pretender to discovery of the laws of nature". So completely dismissive was Tait of metaphysics, that he even placed ideas of perpetual motion ahead of that useless subject. With an attitude like that, Tait was not in a position to uncover any hidden meanings in a metaphysically inspired symbol from Hamilton.

While Tait had very little tolerance for metaphysical arguments, Maxwell on the other hand attends the lectures on metaphysics given by Sir William Hamilton at Edinburgh32 in 1848-49, and displays somewhat more appreciation for the subject. In a letter33 to Litchfield, dated 4 July 1856, Maxwell writes, "I find I get fonder of metaphysics and less of calculation continually and my metaphysics are fast settling into the high style." He considers that his grasp of metaphysics has already exceeded that of many traditional philosophers. And just over ten years later, in 1867, we find him, in a letter34 to Tait, writing, "I have read some metaphysics of various kinds and find it more or less ignorant discussion of mathematical and physical principles, jumbled with a little physiology of the senses. The value of the metaphysics is equal to the mathematical and physical knowledge of the author divided by his confidence in reasoning from the names of things."

The next year, in a letter35 dated April 1868, Maxwell writes, "The practical relation of metaphysics to physics is most intimate." Here he echoes W. R. Hamilton's views and the views of many other metaphysicians. But, he also goes on to clarify his opinion once again, that "Metaphysicians differ from age to age according to the physical doctrines of the age and their personal knowledge of them ... The Edinburgh & the Hamilton differ in their metaphysical power in the direct ratio of their physical knowledge (not the inverse as most people suppose)."

Maxwell, like Tait, didn't have much of a stomach for the kind of metaphysics that came from non- scientists, which often seemed to consist of nothing but a vacuous play on words. He was looking for substance in the metaphysics that directly related to things that could be measured. There were two famous intellectuals both with the same name Sir William Hamilton in his day. One was Sir W. R. Hamilton of Dublin, who was a mathematician, physicist, astronomer, metaphysician, and the inventor of quaternions. And the other Sir William Hamilton of Edinburgh was a rather famous Philospher who taught Logic and Metaphysics at Edinburgh. Maxwell obviously had the opportunity to hear the views of both these gentlemen, and in his mind there was definitely a clear distinction between the type of metaphysics that came from the mathematical sicentist verses that of the pure philosopher. The scientist resonated with Maxwell's own position, but the philosopher was often peddling hot air.

As Maxwell concludes this letter, "I happen to be interested in speculations standing on experimental & mathematical data and reaching beyond the of the senses without passing into that of words and nothing more."

Although declaring more respect for the metaphysics of the Dublin's Hamilton, Maxwell nevertheless was not as ardent a metaphysical advocate as W. R. Hamilton himself. In the review article36 "On Quaternions," published in the December 1873 issue of Nature, Maxwell describes Hamilton's method of presenting the subject of Quaternions, and lodges a minor complaint, "Sir W. R. Hamilton, when treating of the elements of the subject, was apt to become so fascinated by the metaphysical aspects of the method,37 that the mind of his disciple became impressed with the profundity, rather than the

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simplicity of his doctrines." He then goes on to praise the new text just published by Kelland and Tait, Introduction to Quaternions (London, 1873), for bypassing all metaphysics, that was so discouraging to many new students, and getting directly to the of the subject. Metaphysics was fine, but just not in introductory material, and when metaphysical arguments were used, they were better when backed by concepts that had obvious physical meaning.

Given Maxwell's appreciation for metaphysical ideas and keen interest in the subject of quaternions, one wonders then, why he didn't see the metaphysical implications of the shape of symbol which Hamilton chose for his operator. Unlike Tait, Maxwell was in a much better position to grasp the significance. Indeed, the very fact that Maxwell asks Tait about the reasons for the shape, indicates he was certainly reflecting on the choice of symbol. And in 1867 when Maxwell writes Tait on this topic, the horizontal wedge is still the dominant form being used in the published writings on the subject. Here, however, we draw a blank. We are unable to answer this question definitively. An "unknown" of Maxwell's private notebooks, letters, and unpublished papers, went missing after his death, and the collection is conjectured to have been destroyed by a fire that completely burnt down the Maxwell house at Glenlair after his wife's death in 1886.38

This would normally not be such a terrible loss, given that the many ideas scientists develop during their lifetime are usually published in the journals and textbooks they produce anyway, and these documents maintain a permanent record of their achievements. However, Maxwell was discouraged from presenting his ideas in quaternions. And whatever he may have developed, thought about, discovered, or gleened through his reflections on the matter, would more likely be found only in his letters and private notebooks.

In discussing Maxwell's writings, Harman tell us that39, "In November 1870 he wrote to Tait, signalling a keen interest in quaternion ideas, methods and notation...These letters give the first indication of his resumption of work on the Treatise and of an intention to remould its mathematical argument. His correspondence with Tait at this time also provides evidence of his first serious interest in quaternions. He aimed to demonstrate the application of vectors to the mathematics of ."

The "vectors" here, of course, are Hamilton's vectors, the quaternions with no scalar part. Maxwell was becoming serious about quaternions, and thinking about how the theory of electricity and magnetism could be cast using Hamilton's invention. But, there was a problem. The publisher Bartholomew Price, Secretary of the Delegates of the Clarendon Press, Oxford, writes to Maxwell in a letter dated 4 January 1871, "Quaternion Methods and Quaternion Notation are only just beginning to be used in this place: and the exclusive use of them in your book would therefore much curtail its usefulness, so I think you had better always express the analysis in the ordinary Cartesian form, and repeat it when desirable to do so in the Quaternion form."40

Price tells Maxwell "you had better" express things in coordinate notation. There was no negotiation here. Maxwell does exactly what Price tells him in this directive. In the Treatise on Electricity and Magnetism41, he writes everthing in coordinate notation, and only then are some of the main final results repeated again in quaternions. Price's position may be understandable. No publisher wants to produce books that are difficult to sell. And even if a profit is not expected, the publisher wants books that are "useful" to as large a reader audience as possible, because there is still that goodwill and advertizing value that comes from books you've published that are used more frequently. The publisher has to balance the books that are produced at a loss, but keep their name in the public eye, with the fewer more successful books that sell well and bring in the real revenue. Unable to precisely predict which book will be a bestseller, the next best thing is to set the criterion of "usefulness" of the book.

Whatever the motivation, it was shaping the language of physics.

Maxwell was being discouraged from publishing his ideas in the way he might have wanted. He was not free to express himself. And so, for certain, whatever ideas he might have had, resulting from his exploration into the applications of quaternions to electric theory, would remain in his private notebooks, unpublished writings, and letters. Did he then, come to a different understanding of Hamilton's quaternions from Tait? After all, he had the benefit of Metaphysics from both Hamiltons, and already showed that W. R. Hamilton's Operator symbol intrigued him.

Moreover, Maxwell shows a keen appreciation for the importance of symbols in another remark in that

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same December 1873 Nature article "On Quaternions," where he writes42, "The mason's rule teaches us that symbol, as written on paper, is not a , but a mere injunction, commanding us to measure out in a certain direction a vector of a length so many times that of the rule. Without the rule the symbol would have no definite meaning. Thus the rule is the unit of the Quaternion system, while the square reminds us that the right is the unit ." With this aptitude for interpreting the symbolic, displayed here in his speculations on masonic tools, it becomes even harder to understand how he could have missed the implications of Hamilton's symbol.

The missing documents were all from the critical period 1860-79, starting just about the time Maxwell begins to publish on electricity and magnetism, through the period of his explorations in quaternions and the development and writing of his Treatise, up until his death. The first part of Maxwell's life, before 1860, is documented in great detail, in the Life of Maxwell43. But, as Harman writes44, "Campbell declared that 'from this point [1860] onward the interest of Maxwell's life (save things "wherewith the stranger intermeddles not") is chiefly concentrated in his scientific career'." From his 1873 public, but unsigned45, speculations on masonic symbols, we are led to conjecture that that "wherewith the stranger intermeddles not" was Maxwell's private activities and involvement in the Secret order of Freemasons.46

The implication being, Maxwell's interests now were not only metaphysical, but perhaps mystical, magical, and occult, or whatever other description one might give to the "craft" of Freemasonry. The language of his writing also hints at his interests, giving us the famous Maxwell's Demon, as an interpretation of the hidden workings of physical phenomena, and in the conclusion of the December 1873 review article "On Quaternions", he refers to "Quaternions" as that great "spell"47 cast by Hamilton. How seriously one should take these references is unclear. But, one could easily adapt them to construct an alternative explanation of why his private letters went missing, and why Garnett may have chosen not to include some of them in the biography, as they may have revealed things about the Secret Order the publication of which would be in violation of the Order's rules. One could also understand how the burning down of the Maxwell's house conveniently ensured no unfound documents of this type would ever show up unexpectedly later. The only pity is that perhaps some collection of private scientific papers and notebooks, which may have had nothing to do with the Fraternity, may nevertheless have been an unintended casuality of such a protective scheme; but these are speculations.

At the conclusion of the "General Introduction", of The Scientific Letters and Papers of James Clerk Maxwell Vol.I, Harman indicates that of Maxwell's three regular scientific correspondents, Tait, Stokes, and Thomson, the only substantial number of surviving documents in this class are the Tait letters. The main letters missing are the many correspondences between Maxwell and the two famous important contributors to science, Sir George Gabriel Stokes and Sir William Thomson (Lord Kelvin). Stokes was a mathematical physicist who was the leading expert in hydrodynamics48, the very field Maxwell drew analogies from in formulating his Theory of Electricity and Magnetism. While, Thomson was the leader in the field of Heat and Thermodynamics, including the researches into Thermoelectricity, formulating the famous Kelvin Relations, Thomson Heat, and the Thomson Heat equation. Thomson had succeeded in uniting the two thermoelectric effects, the 1822 Seebeck effect and the 1834 Peltier effect, into one general theory, in the years 1848-54, and proceeded to postulate the existence of a third effect now called the Thomson Effect, discovering the latter experimentally in 1854. These were the two key men to whome one would have turned to in Maxwell's time with any ideas that might link thermoelectricity with electromagnetism, and Maxwell was well aquainted with both men. What, if anything, Maxwell may have discussed with them privately on the matter, can only be conjectured. The evidence49, it seems, is no longer in existence to say, one way or the other, if any progress was made to link the two subjects, nor whether even the mere idea of linking the two was discussed at all, although, it is exceedingly difficult to understand how it could not. As with Hamilton's Operator , whatever the architect thought on these subjects are now forever gone with him to the netherworld.50

. What did Maxwell know? On August 28, 1879, just ten weeks before his death, Maxwell sends Tait a very strange letter50a, as if to suggest a hint to his life long friend, who he playfully refered to in the letter as the "Headstone in search of a new Sensation." Maxwell pretends that he is Tait, writing to himself about some new discovery that will solve all his troubles. In the message Maxwell writes the name of the famous saint ALBAN, turns the page and notices, "just opposite the name of the saint another name which I did not recollect having written. Here it is,

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"Here then was the indication, impressed by the saint himself, of the way out of all my toubles. But what could it mean?" He reverses each "letter" of the word NABLA, but not the name. There's apparently no record of anyone ever having written the reverse symbolic letter, , to Hamilton's original differential operator, , back then. Why not? According to Knott, this was the last letter Maxwell wrote to Tait.50a

Did Maxwell just figure out how to write the reverse nabla without actually writing the reverse itself, and wish to communicate this pun to Tait, with whome he shared a secret that could not be written? Or, did Maxwell, knowing he was about to die from his life long illness that had reached the critical point, with only a few weeks left to live, and desperately desiring to tell Tait a secret he'd long since known, but was sworn to secrecy even from Tait, construct this curious tale to send his old best friend a hint, on some simple but essential fact that Tait had completely missed in all his work? As was noted above, Tait complains that he kept bugging Hamilton for more information on , and Hamilton kept promising to supply it, but never did. This led Tait to conclude that Hamilton's work was lost after his death in 1865, because Tait was not aware of any further writing on the subject, yet he believed Hamilton was working diligently on it. But, upon reading Graves biography of Hamilton, Tait comes to the conclusion that Hamilton must not have written anything further after all, because Graves only mentions that Hamilton had "intended" to include the theory of this operator in the last chapter of the Elements which stood unfinished at the time of his death.

Did Hamilton indeed write something more on the subject that did not circulate in public, that perhaps Maxwell nevertheless was able to see? After all, if Tait was such a close friend, and Maxwell had himself independently discovered the theoretical ideas that extended the nabla concept to both left hand and right hand operations, he should have had no problems directly telling Tait of his own idea. In fact, in such an instance, we'd expect Tait to be the first person Maxwell would discuss this concept with. However, if Maxwell did not originate the conceptual extension himself, but was shown a document, perhaps of the private unpublished works of Hamilton, discussing the theoretical extension and its application, he might not be at liberty to discuss this with anyone outside of whatever private society his membership in gave such access to restricted documents. Now, Joly is very careful in picking his words when he tells us that no manuscripts of Hamilton could be found in "the Library at Trinity College," leaving open the possibility that some written works of Hamilton could possibly show up elsewhere, like Hamilton's home. And this suggests that he Joly at least "suspected" there might be such documents somewhere else, even if he didn't know for sure. One could argue that Joly was part of the same inner circle as Maxwell and Hamilton, but that excluded Tait, and he knew of the existance of non-circulating writings of Hamilton, and this knowledge caused him to be very careful when indicating his reference sources in his own 1905 Manual of Quaternions work, compelling him be cautious to make the kind of statements that would be construed as true regardless of any future revelations that might come forth. On the other hand, one could argue that Joly was excluded from that same inner circle, and from conversations with Tait had come to the conclusion that it was highly likely some writings by Hamilton on nabla might exist somewhere else, so was being scientifically careful to state only where he himself had been able to search.

Just before revealing the mirror image nabla to Tait, in the very same last letter, Maxwell subtly explains to Tait, as if by way of an apology for keeping the secret from him for so long, that there are only a finite number of ways things of this universe can be arranged, and once one has seen them all, there's nothing more new to see. His "triple bob major" comment, followed immediately by his reversed nabla, then suggests to us the three major forms of nabla: right, left, and middle. Only the nabla acting to the right appears in the public written scientific literature, and if Tait is seeking a new sensation, he would do well to think along the lines of the other ways things can be arranged. Whether Tait understood the message or not is impossible to say. But in the last sentence of the letter Maxwell teasingly chides Tait for his tendency of going to sleep on hearing anything so profound; a statement that Knott misinterprets as a reference to Maxwell himself, because Knott could not interpret the communication. But, it seems Maxwell may have been very familiar with these other ways to differentiate since the time of his publication of his Treatise on Electricity and Magnetism in 1873. We refer again to the article "On Quaternions," published December 1873, where Maxwell goes into yet another strange elliptical discourse with his readers about men being born with thoughts that "shall always to the right hand, and never to the left,"47 thus subtly explaining to us later why his own Treatise of the same year did not contain the obvious alternative ways to arrange things, even though he hmself may have been aware of these other varieties as early as 1873, and perhaps even

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since he'd quizzed Tait about nabla's shape back in 1867, for it was thought best to leave them out at the time, until the age when men would not be inclined use this knowledge for evil. Thus, Maxwell works only with the nabla acting to the right in his public writings, consistent with everyone else at the time. And while Tait may have been in the dark upto the time of Maxwell's passing, we certainly know he became aware of this issue in the years to follow, at which point he strongly opposed such innovation.

We know that by 1893 McAulay50b revisits this same issue with Tait again, in his attempt to introduce a more general form of the quaternion differential operator, but Tait rebukes him, on grounds that McAulay's innovations were adding to the complexity of an already complicated subject, and it wouldn't help in the ongoing battle between the vector camp and the quaternion alliance. McAulay was on the right track, but missed the simple "major forms" of the differential operator that Maxwell alluded to in his cryptic letter to Tait, and instead had constructed a differential notation that was in some sense too general once again. Both Tait and McAulay were right from their own points of view, but perhaps Maxwell was the most right, for Tait's troubles with the vector camp might have been indeed eliminated by some new sensation from the "triple bob major" of nabla operations, if only Tait was not asleep at the helm. Quaternions might not then have been persecuted and executed like saint ALBAN, by the countervailing masters of mathematical science. But the Nabladists lost their cause in 1913, the year that the will to succeed fell, the headstone organization lost it's organizational head, decapitated in astonishing similitude to the martyred saint ALBAN, as the major supporting "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics" capitulated to vectors and closed its doors for good.50c

Charles Jasper Joly(1864-1906), Irish Mathematician, Astronomer, and Physicist, succeeds W. R. Hamilton in the post of the Astronomer Royal of after Hamilton's death in 1865. Joly edits the 2nd Edition of Hamilton's Elements51, attempting to complete the work along the lines Hamilton had intended, enlarging the work and turning the one volume text into two volumes. Volume 1 is published in 1899, and volume 2 in 1901. He includes an Appendix dealing with Hamilton's Operator, but uses instead of , nevertheless keeping the definition and the spirit of this operator otherwise intact. Later, in 1905 Joly publishes his own work titled A Manual of Quaternions.52 And in this text he shows that he understands the distinction of left-right differentiation.

In Art 5753 he gives the separate definitions for the two, q and q , and tells us that the operator is acting on the variable q in both cases, from the left towards the variable on the right in the first instance, and then from the right towards the variable on the left. Joly has to explicitly tell us this in the text, because it is usual for the operator to be placed on the left of whatever variable that operator acts upon. This is by convention. And normally, q would be interpreted to mean the q was simply multiplying the result of whatever that came out of that operator acting upon on its right, and q would not actually be differentiated at all. Joly then tackles the obvious problem of when to differentiate and when to multiply by using different typesetting fonts. When the variable q was to be the subject of differentiation by the operator, this variable would be written using a Roman Typeface letter, but when being considered a constant, only to be multiplied, this parameter would be written using a Greek Typeface letter. This partially gets around the problem, except, of course, when you need to indicate you're "differenting a constant" !

It is strange that Joly goes to all this trouble to construct a mechanism for the recognition of left-right differentiations, which doesn't even quite work properly, and he never seems to give a thought to the shape of the symbol , that was cast away in favor of , which would have so easily facilitated this very task he was struggling with. Just flip Hamilton's operator symbol left-right, and the problem is solved. No need for all the tricks of typesetting fonts, and having to warn the reader in the text what you're about, because of the fact that if you don't tell your audience your operator is now acting the other way they'll definitely misinterpret what you've written down. There's no visual hint to guide the intuition. You can't just write the math, you've got to write an explanation of what that math means in this context too. How inefficient! How did Joly miss the significance of Hamilton's symbol? We get one clue from his attitude in a brief comment he makes to one of his exercise questions.

In Art 57, Exercise 11, he writes54, "It is sometimes convenient to place the operator to the right of the operand."

To Joly, it was simply a matter of "convenience," nothing more than that. There was nothing

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profound, fundamentally revealing, nor anything considerably material, and nothing terribly significant, about the distinct left-right differentiations. One was at liberty to select the operator acting on the other side guided only by the necessity of the convenience. In this respect, Joly's attitude was very similar to Heaviside's on this same issue. It is possible he may have even been influenced by reading Heaviside views.55

Oliver Heaviside (1850-1925), English Physicist, but also somewhat of a Mathematician, takes apart Hamilton's Quaternions to construct his own version of vectors. Heaviside had four basic problems with quaternions. First, the quaternion didn't represent anything physical, it was just a mathematical abstraction, unlike the vector which was clearly physical. Second, the square of a vector was a negative quantity, but true physical vectors, it was felt by many scientists, ought to have positive squares. Third, the scalar and vector parts of the quaternion were stuck together, but when actually applying quaternions to a physical problem one always had to separate the scalar from the vector part and treat the two as separate quantities in any real application. And fourth, the whole calculus of quaternions was too cumbersome to work with, too complex to understand, and too difficult to apply to most physical problems in general.

Heaviside first encounters quaternions when he tries to read Maxwell's 1873 Treatise on Electricity and Magnetism, and he turns to Prof. Tait's 1867 Elementary Treatise on Quaternions, hoping to get a better grasp of the subject, but as he remarks later in his Electromagnetic Theory, "on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vectorial aspects antiphysical and unnatural, and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalars and vectors, using a very simple vectorial algebra in my papers from 1883 onward."56

Part of the fault can be placed squarely with Hamilton. A quaternion was generally written in terms of the sum of its scalar and vector parts, q = Sq + Vq, or in terms of the product of its and versor parts, q = Tq Uq. In fact, Hamilton introduced six different operators, S, V, K, N, T, U: Scalar, Vector, Conjugate, , Tensor, and Versor; defined for a quaternion, q = w + ix + yj + zk, thus,

Sq = w Vq = ix + jy + kz Kq = Sq - Vq Nq = qKq Tq = +sqrt(qKq) Uq = q/Tq

And all these extra operators added to the complexity of the algebra which would otherwize just have the four rules, +, -, x, /.

This was a major part of the problem.

One has this very beautiful expression for the rotation of a vector, v' = qvq-1. A simple form, that makes use of nothing more than the four rules of arithmetic. With no special constructs, nor any artifices required, to perform the work. Making the algebra at once appealing to the mind, evidently useful, and profoundly elegant, even perhaps astonishingly so.

Then come these altogether arbitrary operators, Sq = q0 and Vq = iq1 + jq2 + kq3, that extract the scalar and vector parts of a quaternion, seemingly imposed on the natural algebra from the outside, like invaders into the realm of logic, extraterritorial actors, who leave no clue as to how they do their work, ruining the simplicity of the algebra, and creating a disjoint impression in the mind of the mathematician, it is no wonder physicists and many mathematicians alike felt uncomfortable with Hamilton's art. How does the vector operator in Vq do its work? Are S and V primitive operators like '+', '-', 'x', and '/' ? Or, are they just convenient shorthand notation for particular sequences of these more primitive four rules? The distinction is critical.

If S and V are just conventient shorthand, then there is no problem. One can always replace these operators by the sequences of steps in the regular four rule operations they represent. In this case, the operators S and V are not a necessary part of the algebra, and the algebra retains the simplicity, wholesomeness, and undivided structure, of all simple number , while introducing much more

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power in the representation of complex operations.

But if S and V are necessarily primitive operators, that cannot be further reduced to the usual four rules, and we must nevertheless work out our calculus with them, then we've really got two separate algebras, an algebra of scalars and another algebra of vectors, that are merely being hung together, as one perhaps shaky structure, for no immediately apparent good reason.

From the manner in which Hamilton prepared and presented the Quaternion Theory, it often appeared as if the latter were really true. Indeed, Heaviside, and many others, really felt that the algebra of vectors and the algebra of scalars were obviously two separate and distinct algebras masquerading as one under this peculiar four dimensional structure Hamilton had called quaternion. And all the actual links that existed between scalars and vectors were seen by them as hinderances to the usefulness of the art. It often seemed as if many of the actual steps in the algebraic manipulation that occurred in a calculation involved the effort to separate S from V, so that one could get at and deal with the vector part one needed for physical application.

Hamilton himself had begun the process of dismantling the quaternion with his very definition, q = Sq + Vq, and use of this expression from the outset. As Heaviside saw it, this was clear evidence that quaternions represented two distinct algebras stuck together. The prominant role these operators, S and V, played in the calculus demonstrated to many observers that quaternions as a coherent system of four degrees was merely marking time, only three degrees were needed for space representations, and the fourth scalar was simply spurious. One always seemed to be chopping up the quaternion into these two distinct parts, Sq, and, Vq, and for some unknown reason they were required to be put back together again at the end of calculations. Why, just to say you have a quaternion?

Well, would have none of it. He saw no reason to bother himself with the cumbersome quaternion calculus. And, convinced that it a was fatally flawed theory that needed replacing, he set about to finish the job, which Hamilton appears to have unwittingly started, of dismantling the structure, by permanently detaching the scalar and vector parts from each other in all operations. To accomplish this, he first redefined the squares of the unit directions to be positive, so

i2 = -1, j2 = -1, k2 = -1, became, i2 = +1, j2 = +1, k2 = +1.

This allowed all vectors to have positive squares instead of the "unnatural" negative squares. Then, he replaced the single quaternion multiplication by two others, a vector product and a scalar product. But, he kept to some of Hamilton's notation57, so that the new "vector product" was written, Vab, while the new "scalar product" was written, ab. These would later become popularly known as the "" and the "," names given to these operations by Prof. Gibbs of America's Yale University.

In proceeding to develop his new vectorial system, Heaviside then encounters his first real "ambiguity." Having defined two types of multiplication, a scalar product and a vector product, he attempts to define corresponding divisions. Hamilton's quaternion obeyed all the four rules; add, subtract, multiply, and divide. So, if one is going to replace the quaternion product with two alternative products, naturally, we've got two divisions also to consider. Corresponding to the scalar product, ab, one could simply define scalar , a/b. But, for the vector product, Vab, which would we pick for the vector division, Va/b = Vab-1, or, Va/b = Vb-1a ?

This problem appears trivial. But, it is not. It gets to the very heart of the issue surrounding that same quaternion polarity.

In his June 1891 article, "On the , stresses and fluxes of energy in the electromagnetic field," where he gives a brief exposition of his new , Oliver Heaviside writes58, "As A/B is a scalar product, so in harmony therewith, there is the vector product VA/B. Since VAB = -VBA, it is now necessary to make a convention as to whether the denominator comes first or last in VA/B. Say -1 therefore, VAB . Its tensor is V0A/B = A/B sin(A^B)...(14)"

In quaternions, this problem shows up much more clearly. We can either divide from the left or right, q-1p, or , pq-1, and these generally yield two very different results because quaternions don't commute. Which one of these should we pick for the definition of "division", p/q ? There is an

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ambiguity. But, that ambiguity only exists because we want a single answer, while the truth is that there are two equally correct answers. Rather than pick only one, and ignore the other, we can easily define these two divisions using both the forward slash '/' and the backslash '\', so that, p/q = pq-1, and, q\p = q-1p. This gives equal prominence to both ways to define division, acknowledging the correctness and importance of both left and right divides.59

But, this was not the tact Heaviside displayed in his approach to the similar problem in vectors. While he is forced to recognise that there is a "choice" between two equally valid forms for division, he cannot see greater significance beyond this, that is implied by the mere existence of the ambiguity. For Heaviside, this was just a matter of "convention." He simply picks one form, discards the other, and proceeds to develop his vector algebra accordingly. Heaviside's "convention" was Joly's "convenience." Both men recognised the left-right ambiguity inherent in the calculus at various places and thought nothing more significant than an arbitrary selection of one was required to resolve the matter.

Many mathematicians and physicists showed the same blind spot right here at these issues of ambiguity. You have two ways to divide, two ways to multiply, two ways to operate on a variable. For the most part, academicians simply ignored one alternative and proceeded to develop arguments with the chosen action, as if the one cast away did not matter, and could not play a significant role in the development of the unfolding theory.

Heaviside could possibly be partially excused, because the way he had defined things, the difference of division from the left and right was simply a matter of an overall minus sign. The parameters in the vector product anti-commute, but the parmeters in a quaternion product generally non-commute. So, to him, it was, Va/b = +Vab-1, or, Va/b = -Vab-1 ?

It is easy to miss the importance of this subtle distinction, when only a '-' sign differentiates the two forms. This is because one gets the feeling that the alternative option can always be expressed by simply putting that '-' sign back in front of the chosen definition, and one really hasn't lost anything at all. In quaternions, the distinction is much greater. But, in vectors, the problem seemed to have remarkably faded away. This left-right ambiguity in the division operation of , extends over to the differentiation in quaternion analysis, since one would like to take a function, p (q), make a small change in the independent variable, dq, examine how the function changes, dp, and "divide" the function change by the variable change, take the limit, and establish the derivative;

dp/dq = dp . 1/dq -or- dq\dp = 1/dq . dp

Only now, there are "two" ways to divide, dp/dq and dq\dp, in quaternions. Which should we pick? The existence of distinct left and right division operations clearly implies the existence of distinct left and right differentiations. But, when we cast away one divide, we also tend to cast away the corresponding alternative derivative result along with it.

Heaviside had gotten rid a good many things. Hamilton's many operators, S, V, K, N, T, U, were reduced to just V. The "mathematically abstract" (i,j,k) basis definitions, now became the "physically representative" (i,j,k) with positive squares. The permanent separation of scalars and vectors was acomplished by replacing the single quaternion product by the separate "scalar product" and "vector product". Left-right ambiguity was resolved by "convention." And the complicated quaternion calculus was discarded, while vectors took on a comparatively more pleasing appearence that made them easily marketable to scientists and engineers alike. It wasn't that vectors were superior to quaternions, the truth of the situation was quite the contrary, but Hamilton's presentation suffered from many defects.

Hamilton had dressed the Quaternion up in rags and paraded him about the streets, and no one could recognise the King. Underneath the un-princely garb, stood the monarch of algebra, unseen by the multitudes of mathematical passersby, clothed in peasants garb, a patchwork quilt of shaggy operators, stuck together in unsightly manner, quaternions were doomed to leave an unfavorable imprint.

When it came to the representation of a space rotation, qvq-1, the form of the quaternion expression is found to be remarkably reminiscent of the actual form physical actions take when producing a rotation. We find the algebra requires a simultaneous action from the left and from the right, equal

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but opposite actions, q/1 and 1/q, both acting on the object in the , v, that is the subject of the rotation. In the real world, of physical phenomena, what do we find? Two equal and opposite forces, F and -F, acting simultaneously from opposite sides of a material object with some of , I, producing a , and causing the object to rotate about its center of . The parallels could not be any more striking. Here was algebra, modeling physical phenomena, with piercing accuracy. Truth standing up on its own in nature's most natural symbolic garb, with no attempt nor any effort required from the mathematician to patch it up and make it fit the observed reality.

But, the clear light that shone so brightly on the rotation expression, was soon dimmed by the dusty, murky, and drab forms that arose out of the six dark knights of Hamilton's confounded calculus. What was he thinking?

Just about the time he discovered a new way to represent ordinary complex , using ordered pairs (x,y), instead of the familiar x + iy notation, leading to his development of the Algebra of Couples, Hamilton had an inspirational idea that later became the focus of his essay on Algebra as the Science of Pure Time. His idea was very simple, algebra should mimic physical reality and be based on it, instead of just being founded on symbolic logic. His arguments lost out to the mathematicians who followed Boole in regarding logic as the very foundation of all mathematics, and were happy to construct any kind of abstraction, however useful it might be, once such construction was consistent and sound in the laws of symbolic logic. Hamilton disagreed with that view, for he felt that algebra, in some sense, yet not too clearly specified, must somehow be used to describe the sequences of actions and patterns of events as found in the real world, and the algebraic operations should therefore have their parallels in the physical world of known phenomena.

We may state this principle simply as "Every algebraic operation must correspond to some physical action." Although, Hamilton tried to define the relationship between symbolic representation and physical phenomena in a somewhat different way than this, using sequences in time, it seems in his rather more roundabout way he was simply trying to state this principle. Obviously, physical actions take place in time, one after the other, and the algebraic operations that describe the changes from one state to the next must then be modeling this progression of acts in time. But, it is the physical actions that are the focus of this process. This is what an algebraic operation should encapsulate. Clearly, quaternions hit a high note in this regard with the rotation expression.

On the 11th September 1846, in a letter to his father Rev. James Hamilton, Sir William Rowan Hamilton presents his view on this idea again60, "Indeed I am profoundly impressed with the conviction that if the intuition or conception of time be put out of view, as foreign to Algebra, then Algebra itself must cease to be regarded as a science. It must either descend into the rank of an art, which has for its province to supply convenient rules of calculation; or pass, by what to some minds may seem an ascent, into the category of language, having no independent truth, but only at most a certain coherency or elegance. The symbols will then become, what many now account them to be, the all-in-all of algebra: the analogy to geometry will disappear: the signs will have no longer a reference to things or thoughts signified by them."

But, the high note that was struck with the rotation operation, was not always repeated with the rest of the expressions in the calculus. Some things were indeed beautiful and simple, but very many others were obscure and understood only by the experts in the field, because the expressions were often encumbered by an arbitrary kind of complexity on which the intuition faltered. Much of this obscurity was caused by the seemingly arbitrary nature of the operators, S, V, K, N, T, U, that were too oft used without any deliberate attempt to reconcile them with the sequences of more primitive operations they might represent. There was therefore no feeling for how exactly these operators were to be related to something physical.

Hamilton lost sight of his own intuitive insight, which originally caused him to give the world his Algebra as the Science of Pure Time. After this initial inspiration, he seems to have lost the ability to focus on the principle as he later developed his quaternion calculus. And his calculus seemed to descend towards that very abstract morass of symbolic logic constructions he had himself complained about. Fortunately, much of the apparent descent was only an illusion, caused in part by the cumbersome nature of the notation used to develop the art, and the real beauty and simplicity lay beneath the towering mass of often obfuscating symbolism. But equally unfortunately, few had the patience or the resolution to trek through Hamilton's tomes to extract the light of his revelations, and aquire the understandings that were so inspirational to Hamilton himself. And even Hamilton missed the opportunity to develop and emphasize those key expressions that would have thrown more light on exactly what those often used six major operators might mean in a physical context.

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This power of quaternions to suggest to the mind a visual picture that closely corresponds to some equivalent physical phenomena in the real world has been refered to by JD61 as "Hamilton Visualization" [HV]. And it seems that W. R. Hamilton himself, for the most part, was able to see past his own shorthand and obtain that powerful visualization without the benefit of seeing the expressions in their naked form. But most readers of Hamilton's Quaternions could not see past the cumbersome notation to attain that same grasp.

Josiah Willard Gibbs (1839-1903), American Mathematical Physicist and Theoretical Physical Chemist, Professor at Yale University, co-conspirator with Heaviside in pushing the "new" vectors on the , develops his own version of vectors independently of Heaviside, and the two men only become aware of each others work after their own separate developments have been essentially completed.

Heaviside says62 - "upto 1888 I imagined that I was the only one doing vectorial work on positive physical principles; but then I received a copy of Prof. Gibbs's Vector Analysis (unpublished, 1881-4). This was a sort of condensed synopsis of a treatise. Though different in appearance, it was essentially the same vectorial algebra and analysis to which I had been led. That is, it was pure vectorial algebra, and the method of treating by the operations potential, curl, divergence and slope was practically the same. Not liking Prof. Gibbs's notation so well as my own, I did not find it desirable to make any change, but have gone on in the old way."

Heaviside may not have liked Prof. Gibbs' notation, but the rest of the scientific community took to it like a duck to water. And today, the symbols, '.' and 'x' and , and terms, "dot product" and "cross product" and "del", form the language of vector calculus, and Heaviside's notation is no longer in use; although ocassionally one hears the terms "scalar product" and "vector product" and "nabla".

The original four unit numbers from Hamilton's quaternions, (1,i,j,k), with their single product law, now required four different types of products; (1) the usual product of two real numbers, (2) product of and vector, (3) dot product, and (4) cross product. Thus was the Heaviside-Gibbs vector algebra defined;

1,i,j,k

i.i = 1, j.j = 1, k.k = 1, "dot" products "a.b"

i.j = j.i = 0, j.k = k.j = 0, k.i = i.k = 0,

ixi = 0, jxj = 0, kxk = 0, "cross" products "axb"

ixj = -jxi = k, jxk = -kxj = i, kxi = -ixk = j,

1i = i1 = i, 1j = j1 = j, 1k = k1 = k "ordinary products za, zy"

1(1) = 1

This was all necessary to permanently separate the "scalar" from the "vector". And having achieved this, the new underlying basis for the Hamilton Operator (id/dx + jd/dy + kd/dz) now changed the actual meaning of this operator too. Now, it was not just the explicit polarity hint in the shape of the symbol, , that was gone, which some may argue was only a superficial alteration, that had no real impact on the algebra itself, but gone were the links between scalar and vector, and gone was the wholesome coherent algebraic structure that made quaternions one of the four unique normed division algebras.

The new , and its modified ijk-basis, made it possible to just take the "dot product" to obtain the "divergence", div(), and just take the "cross product" to obtain the "circulation", curl(); algebraic expressions that were previously embedded in Hamilton's calculus in a way that must have seemed to physicists an unnatural forced coexistence. With caution6 we can write these expressions in both the old and new notations to compare them;

= -1/2.( A + A ) = div(A)

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. . A . . x A = +1/2.( A - A ) = curl(A)

What we see, is that the new algebra was really subtly incorporating equal combinations of the left and right actions of Hamilton's operator without explicitly saying so. It now hid the "mechanism" from us, by which these operations were achieved. We can look at the curl in Hamilton's language, 1/2.( A - A ), and immediately get that idea of two equal but opposite forces, F and -F, acting together to produce the torque that gives rise to the rotation.

If we recall the rotation of a vector, qvq-1, using the quaternion, q, and now let, q = (1 + )1/2, so that the inverse, acting the opposite way, from the right towards the left instead, in addition to being also inverted, becomes, q-1 = (1 + )-1/2, then with the assignment, v = A, and using the approximate expansions, (1 + x)1/2 = (1 + 1/2.x + ...), and (1 + x)-1/2 = (1 - 1/2.x + ...), we have;

(1 + )1/2A( + 1)-1/2 = (1 + 1/2. )A(1 - 1/2. ) = (A + 1/2. A - 1/2.A + ...) = (A + curl(A) + ... )

We see immediately a confirmation of that idea of the curl of a vector as being the consequence of the rotary transformation on the vector. Indeed, "curl(A)" was often written "rot(A)" by various authors in the past. The curl describes the transformation that changes A into qAq-1, the "before" and "after" states of the given vector quantity.

curl(A) = (1 + )1/2A( + 1)-1/2 - A

One can easily put a small scale measure, h, into this expression, q = (1 + h )1/2, so that the rotation quaternion is a function of that scale parameter, q = q(h), and consider the more appropriate limit, curl(A) = (qAq-1 - A)/h, as h --> 0, to establish a definition for the curl as an instantaneous measure of the rotation. Compare this with the operation that rotates any vector within the defined by the inital and final vector states. If a and b are the initial and final vectors before and after a rotation, then the quaternion operator that describes this rotation is given by, q = (b/a)1/2, for then b = qaq-1, and q is the of the ratio of the vectors. This particular rotation is restricted to turning about the axis that is normal to the plane defined by the initial and final vectors, and is therefore a special case of the more general types of quaternion rotations that can turn a into b. 62a.

Moreover, if the operations, v, and, v , are considered representative of the forms nabla to-the- right, and, nabla to-the-left, respectively, then these expressions with the square root of nabla, acting from both sides simultaneously, could be considered the forms of nabla that act to-the-center. The modern vector analysis of Heaviside-Gibbs can then be seen as utilizing those nabla to-the- center operation ideas, extracted from the complicated quaternion calculus, and written symbolically, x v and . v, as though they were some kind of nabla to-the-right operations, whereas in fact the new forms are really algebraically 'central' operations, and neither proper left nor proper right handed expressions at all, despite appearances when written.

Of course, given that the vertical wedge, , is indeed symmetrical about the vertical, with no left hand nor any right hand suggestion being implicated in the symbol's shape, and considering that this is entirely consistent with its modern use effectively as a nabla to-the-center operator, one could argue that this was the real reason why the handed form, , was finally turned up to the vertical. This idea would have alot of merit, if only for the fact that nothing remotely suggesting this has ever appeared in the public scientific literature anywhere before.

It may seem that the new vector algebra was simply a shorthand way of writing these left-right quaternion expressions. But, the effect of the changes went deeper than this. One cannot really just

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combine left and right actions from Hamilton's system to make it equivalent to the new vector algebra, because the structures are so different. What we see here, when using the left-right combinations to exhibit the form of the operations in quaternions, is partly the impact of that "first" thing lost when mathematicians depreciated the value of the symbol, , by turning it around 90o, thus returning the important "polarity" to the realm of obscurity.

Correlation63. When the physicists changed the definition of the unit basis (i,j,k) underlying vectors, the "second" thing that was lost was the inherent correlation between the measures of scalar and vector quantities. A certain degree of freedom was introduced into the algebra of vectors that did not exist in the quaternion algebra. Quaternions tied together scalar and vector quantities in such a way that the relative sign of the scalar and vector components fixed the relationship between the sign of the scalar divergence and the direction of the rotary movement. Heaviside-Gibbs vector algebra unlinked the scalar and vector parts and consequently destroyed this particular fragment of information that would otherwize persist throughout the algebraic manipulations. And like the propagation of an experimental measurement error, through many steps of a calculation, would limit the accuracy of the computed result, this little detail propagated a rather substantial difference of some non-trivial significance.

The result of which was that quaternion and vector theorists totally missed the chance to recognise the causal connection between thermoelectric and electromagnetic effects, missing the opportunity to construct a single set of electric equations to join these two disparate subjects together. For the scalar "heat" and the vector "electric field" are naturally linked by the single conception provided by the quaternion, but in new Heaviside-Gibbs vectors there is no guidance for a scalar term to combine with a vector to establish such a fixed relationship. Heaviside and Gibbs broke the link, removing the correlation bewteen the scalar and vector quantities, because they felt it was more of a hinderance than a help in an algebra that was required to model physical phenomena.

In metaphysics the "Law of Cause and Effect" declares that every effect must have a cause, everything in the universe is correlated, and there is no such thing as chance, and nothing is ever really random, regardless of any appearances to the contrary that might exist. We use equations in physical science to model this cause and effect relationship, like, A = B + C. Such an equation tells us that A is perfectly "correlated" with the sum of B and C. And whenever B and C undergo changes, the parameter A will change correspondingly by a known amount, equal to the sum of the changes in the other two parameters.

Some observers make the distinction between correlations that occur at "the same moment" in time and those correlations that occur "across an interval" of time. The former, they declare, cannot represent a cause and effect relationship, because the cause must precede the effect in time. Indeed, this argument is used to prove that Maxwell's Equations cannot describe the cause and effect between the fields. The equations declare that the time-rate-of-change of the electric field, E, at a given moment in time, is correlated with the presence of a space-rate-of-change circulation in the magnetic field, B, but that does not mean that the change in the E-field is the cause of the spatial variation found in the B-field.

In fact, the one cannot cause the other, because both rate-of-change parameters are being measured at the same moment in time. All Maxwell's Equations can tell us here, is that if we look, find, and measure the change in the E-field over some small interval of time, then were we to have simultaneously looked for a B-field, we would have also found a magnetic field existed with a spatial variation consistent with the declaration of correlation in the equations. Neither does the E-field changes cause the B-field ones, nor do the B-field changes cause the E-field ones. All causes really come from the true sources of the fields, the existence of charges and currents, that were present at some earlier moment in time. And it is the influence of these original sources that caused the changes in "both" the E-field and the B-field, only that it happens in such an ordered manner that they show that perfect correlation as described by Maxwell's Equations. This appears all well and good, but there is subtle a flaw in the argument.

The flaw was pointed out in 1748, by the Philosopher David Hume, in his essay64, "An Enquiry Concerning Human Understanding," in which Hume examines that very idea of a "necessary connection" implicit in our understanding of the cause and effect relationship.

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The concept of "Cause and effect", says Hume, is an anthropomorphic extension of the idea of human "will" to the of inanimate objects and lifeless phenomena. All we are really justified in asserting, if we are to be truly objective, is that we observe a pattern, and similarities to other patterns, of sequences of events that occur in time. The "contiguous" nature of events B and A, where the former follows the latter in time, gives rise to their association, and when we observe similar events B' and A' occur again at some other time, the former following the latter once again, being again closely separated in time, we formulate the idea of things like A being the "cause" of things like B. The "pattern," the "similarites," and the "sequences" are objective, because they are observables. But, the "idea" of "A causing B" is not objective, and has no part in objective science.

It is because man, by his own personal experience, knows that "he" can "cause" things to happen, by the exercise of his "will", that the "idea" of "cause and effect" has emerged. But, the inanimate objects have no "mind." ! The lifeless phemonena are not the direction of any "will." So, there can be no "Cause and effect" in nature. Hume argues that all cause and effect explanations must be replaced by the sequential patterns of observables, with a certain temporal contiguity, and the "non- observable" ideas like "cause and effect," "by influence," "force", "induction", etc.. should all be eliminated, because they are nothing but supertitions. Being subjective ideas they have no part in objective science.

But, scientists could not get rid of these ideas of 'cause and effect', of 'force', 'by influence', and 'induction', etc..because they give one a 'feeling' of comphrension. Humans "need" to feel that there is a "reason why" things happen. But "reason" is of the "mind" the "heart" and the "psyche." It is entirely of human value. Thus, science maintains its superstitions, and much of science is anthropocentrism.

Well, anthropocentric or not, scientific observers still insist on employing descriptions of cause and effect to the world of material objects and lifeless phenomena. But, here is the problem. Suppose we agree that it's not the changes in the fields that cause other fields to change, but it is the presence of electric charges and currents, and their variations at an earlier moment that really cause the fields and their modifications to appear. Then, we have to ask, how did those charges get there, and how did the currents flow in the first place. Did another set of electric and magnetic fields influence the creation and motion of those charges, so that they took up the precise static and dynamic profiles required to generate the later observed fields? If so, isn't it really the first set of fields that is the cause of the second set of fields? And the charges and currents really have nothing to do with the cause. They are simply intermediate steps in a long chain of actions that form a neverending sequence. The real cause is unknown. That is, as long as no human investigator is part of the loop.

But, if we ask again, what caused the charges to come into existence? Did a human observer rub his hand on some dielectric material and create that static charge by ? What caused the currents to flow in the first place? Did a scientific experimenter connect an electric circuit to a battery and flip on the switch to initiate that dynamic charge flow? As long as we can identify that early moment where the charges and currents came into existence because of the action of someone's "will", there is no problem with determining the real cause of the ultimate fields and their changes observed and measured at the later moment. This is because, the "will" can cause things to happen. But, when just tracing changes of material phenomena through time sequences of events, all we can really state is that there are correlations between things. There are correlations between observable parameters inspected at the same moment of time, and there are correlations between parameters inspected across intervals of time. Neither type of correlation can actually claim the title of cause and effect, because that demands the intervention of the will of a living, thinking, conscious being to take action and command the first cause.

Now, metaphysically speaking, the "Law of Cause and Effect", is really the statement that nothing happens by "chance". And sometimes this is stated as the "Principle of Law and Order," because of the intricate problems with the ideas of cause and effect itself. What the metaphysical law is really concerned with is the statement that there is order in the universe. That everything works according to precise laws and formulas. That there is nothing that happens for "no reason at all." Of course, one can debate these things. But, the general assertion is that correlations, between parameters that describe observables, exist. These correlations are described by mathematical equations. The equations then illustrate the law of cause and effect at work. To pick out and cast away those correlations that define parameters at the same time, because they can't define cause and effect, is ok. But, you must be consistent and throw out those correlations that occur over time intervals that you can't prove are causal too. That means, all acts and events that occur outside the mind, not directed by a will, have to be considered acausal correlations. Still, there is order in the universe. And we use the term cause and effect to describe that order. Because rather than try to trace the true

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ultimate cause, once we have any known correlation between two parameters, we can "predict" or "say what will happen" to the other parameter when we cause a change, however that may be accomplished, in one of them. By making our measurements of one parameter in this room, we can say what the experimenter in the room next door will be measuring in the other parameter, at the same moment. And that's all we need to convince ourselves that there is law and order in the universe.

. The changes to Hamilton's Operator, , thus diminished the carefully balanced delicate significance of two basic metaphysical laws, "The Law of Polarity," and "The Law of Cause and Effect," that were originally encapsulated by Hamilton's ingenious definition.

The changes to Hamilton's Operator, , thus diminished the carefully balanced delicate significance of two basic metaphysical laws, "The Law of Polarity," and "The Law of Cause and Effect," that were encapsulated originally in his definition.

We should like to emphasise that it was not the laws themselves so much so as the significance of the laws that were diminished. The Law of Polarity can still be seen implicated, although we use the vertical wedge, given that the algebraic structure, after all, practically retains the left-hand verses right-hand distinction. But, the left-right property distinction is one more step removed from our immediate awareness, and has slipped back down towards our subconscious. The great mental effort Hamilton made, to extract quaternions out of the depths of mind, and make a subtle and hidden verity explicit and obvious instead, was thereby partially being undermined.

Turning the symbol around didn't prevent Joly from recognizing the existence of left and right differentiations, which he does in Art.57 of his 1905 text, A Manual of Quaternions. But, the attitude of Joly, that it was simply a matter of "convenience" which way we differentiate, shows that by then the importance of the two distinct left and right operations, and especially the consequental implications of their simultaneous use, did not register in his mind. The significance of the polarity eluded him. Had he the benefit of a few hours meditating on Hamilton's original symbol, he might have understood the significance. But, by 1905 that old symbol has been dead for over 35 years already, and whatever usefulness Hamilton may have thought his symbol might have had in invoking associative ideas, and extracting truth from the dephts of the subconscious, all by the "bare inspection" of this crafty geometric form, was long lost on scientists, mathematicians, and philosophers of the times, who now spent most of their time working and thinking with the wrong symbol fixed in their mind. As we show, it is certainly not just a matter of convenience. 65

It was not just an algebraic matter that could be relegated to simple notation housekeeping either, whether we write AB or BA. Reversing the order, of either the algebric product, or the differential product, implicated different corresponding physical actions were being represented by the algebraic operations. And if nature allowed the one physical action, it certainly required the other opposite physical action. The description of phenomena could only be complete with the inclusion of both poles in any analysis alleging to represent nature. It is only natural, therefore, to indicate the distinction with the use of a polar symbol, one that is not symmetrical. Nature, after all, has two poles, and both are universally present, such is the claim of the metaphysical Law of Polarity.

Changing the definitions of the bases (i,j,k) to effectively separate the scalar from the vector parts of the quaternion, didn't prevent scientists from using scalar and vector quantities together again in their equations to represent the law and order of nature. It was not impossible to express the Law of Cause and Effect by the modern vectors. But, the significance of the Law of Cause and Effect inherent in quaternions was that the quaternion algebra often required a scalar and vector to be linked, whereas in modern vector analysis the scientist could always easily elect to establish this relationship or not. Whether the freedom granted by the modern vectors is indeed a benefit depends on whether the phenomena of nature being represented allows the same freedom in its working.

It seems that nature is far more "restrictive" in its actual activities than the scientists new algebra, and from the experimental thermo-electric evidence it appears that the precise form of the restriction in electric phenomena may be exactly expressed thru that subtle link between the scalar and vector

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parts of quaternions when both the left-hand and the right-hand are properly included in the description of nature's art.66

Conclusion. With suppressed polarity, and destroyed correlation, the the original nabla of Hamilton became a somewhat disabled vector differential operator , inhibiting the intuition from clearly percieving what natural forms representative expressions should take, and lacking the power to implicate the intimate connection between certain related scalar and vector parameters. This is not to say, however, that the Heaviside-Gibbs vector algebra did not have its merits, for it separated and simplified scalars and vectors to the point that certain calculations did become easier, and it again facilitated some applications of vectors to physical science, and made some other results easier to understand. But this all came at a price, that yet other beautifully simple relationships and connections, and especially their significance, simply got lost in the process.

Extension to 4-d. In our work, we resurrect Hamilton's original differential operator, and extend his three-dimensional form to four , while otherwize keeping all the richness of his original conception intact.

The 4-d version of the operator we write67, d/dr = @/@ct + i@/@x + j@/@y + k@/@z, where the scalar axis is identified with time. And in applying this operator to a variable parameter, A, we differentiate to-the-right, d/dr->A, and differentiate to-the-left, A<-d/dr, using the arrows, '->' and '<-', to indicate the directions of operation similar to the 3-d operators, and , of Hamilton. Our differentiation to-the-right, then involves a right-hand rotary movement. While our differentiation to- the-left, involves a left-hand rotary movement. And while Hamilton's symbols also look somewhat like arrows, which can lead to some confusion since our arrows point the other way, if the reader will remember the 'dot' and 'bar' analysis given above for the horizontal wedge, recall Hamilton's symbol refers to 3-d vectors only, and remind himself that the arrows are entirely different things meant to indicate only how "any" operator acts, and so are not themselves differential objects of any sort, all confusion should thereby be eliminated from his mind.

NOTES & REFERENCES:

1. W. R. Hamilton's 1843 paper - "On a new species of Imaginary Quantities connected with the Theory of Quaternions" [communicated November 13, 1843]: Irish Academy Proceedings, II, 1844, pp.424-434. [ The discovery had been made on October 16, 1843], see also History note 1843.

2. Paper communicated to the Royal Irish Academy (July 20, 1846), published in the proceedings, Vol.iii, p.291. See also History note 1846.

There are a possible total of five sources for Hamilton's own writing on his differential operator:

[1]: Southampton Paper (unknown date; 1846 ?); missing paper, where he supposedly used [2]: July 20, 1846, Proc. of the Royal Irish Academy, Vol.iii, p.291 (???); he used [3]: Phil. Mag. 1847, Vol.iii, p.291; he used [4]: Art.620, of Lectures on Quaternions, 1853; he used

[5]: Art.418, of Elements of Quaternions, 1866; where appears disguised as Da.

I have a small problem with the citation [2], which is taken from Joly. I have not seen this reference, and it seems strange that the "Vol.iii, p.291" should be identical to that for the Phil. Mag. 1947 paper, which I have seen, where this symbol appears in the article "On Quaternions; or on a New System of Imaginaries in Algebra", by Sir William Rowan Hamilton, pp.278-293, and which itself is just one of a sequence of articles Hamilton wrote for the Magazine over several years, all with the same title, being considered by him continuations on the same topic. So, I think there must be an error in Joly's citation, but I have no access to the R.I.A Proceedings to check up on this. It is reported by Joly that [3] is just a repeat of what is in [2]. The only real references then being [3] and [4], since [5] doesn't refer to the symbol, while [1] is missing and was never published.

[ See also Earlist Uses of Symbols of Calculus ]

.

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3. We say "mathematicians," but the academicians involved with the development of quaternion and vector mathematics were often both mathematicians and physical scientists, playing different roles depending on circumstance. However, the decision to change the symbol Hamilton used for differential operator was a purely mathematical one, i.e. it had nothing to do with "physical ideas" at the time, in contrast to the other changes that were made later. Acting in their capacity as mathematicians these academicians altered the symbol, their exact motivation is still unclear, but it seems that the vertical wedge may have been easier to write, and some element of fashion may have played a role. The chief culprits here are Peter G. Tait, and Charles J. Joly, although, undoubtedly, the influence of other mathematicians were involved. We should remark here too, that Hamilton himself considered this vertical wedge, and even used it in his first paper on the subject, the so called "Southampton Paper", which paper through error was never actually presented and never published, after which he changed his mind, with further reflection on the choice of symbol, and opted thereafter for the same horizontal wedge that actually appears in all his published writings.

4. Of the various names proposed over time: "nabla", "delta", "del", "atled", etc... we have chosen to use the popular ones "nabla" and "del". Sir W. R. Hamilton's own name for his operator was "characteristic of operation", which is a bit long winded, and which was more of a general name he applied to various other operators upon occasion. But, this was the only name he actually used when he wrote on . And to be accurate, one should point out 'del' was only ever used to refer to the vertical wedge , since this name came much later, by which time the symbol and its operator definition had undergone several changes.

Hague tells us the name 'del' comes from Gibbs, "The differential operator was introduced by Sir William Rowan Hamilton and developed by P. G. Tait; it is of central importance in many three-dimensional physical problems. The symbol was originally named 'nabla' after a harp-like ancient Assyrian musical instrument of similar shape; it is now usual to adopt the term 'del' introduced by J. Willard Gibbs." [ pg.41, Ch.4, The Operator and Its Uses, in the text, "An Introduction to Vector Analysis", by B. Hague, Revised by D. Martin, 1977 reprint, London: Chapman and Hall, 121 pages, ISBN 0-412-20730-3; (first published in 1939; 6th edition in 1970)]

. 5. We say "physicists," because the decision to change the definition of the vector basis (i,j,k) underlying quaternions came from physical scientists, in particular the English Physicist Oliver Heaviside, and the American Theoretical Physical Chemist . But the latter was quite an established mathematician and physicist as well, and Heaviside invented the Heaviside Operational Calculus, a largely mathematical theory, and is also remembered for the Heaviside Step Function. So, the same individuals played the role of both physicists and mathematicians at different times. Nevertheless, it was the "physical concepts" that motivated the change in the definition here, thus we emphasise the fact that these versitile academicians were acting in their capacity as physicists in this context, especially since the ideas of many other physicists influenced this decision.

. 6. See Appendix I for discussion of this "trick" and appropriate "warnings". [ TO BE IMPLEMENTED ... ]

. 7. For modern cross product, vectors anticommute, A x B = - B x A. But for quaternions, vectors generally non-commute, AB = -A.B + A x B, and BA = -A.B - A x B, so AB <> BA, and AB <> -BA either.

. 8. Hamilton, W.R., "Elements of Quaternions", 1866, edited by his son W. E. Hamilton. See History note 1866.

. 9. See also the Historical note 1846. One wonders if Hamilton had any private papers not in the Library at Trinity College, that may have contained his ideas here. Joly is very careful to specify that nothing could be found in this particular Library, leaving open the possibility that such papers might have been located elsewhere, like Hamilton's home, as was conjectured in the similar case of J. C. Maxwell's later missing papers. There is a possible hint here that Joly may have had reason himself to suspect the existance of further notes or writings by Hamilton on the topic. Notes that he could not refer to, either because he was constrained from doing so, or because he wasn't actually sure they existed, hence he was careful not to commit himself to a stronger statement than this about the non-existence of any further such writings. P. G. Tait also felt that Hamilton must have gone further in the development of this operator, and conjectured that any further work must have therefore been lost. But Tait, after reading Graves' Life of Sir William Rowan Hamilton, that appears after Hamilton's death, in which Hamilton is stated as declaring he intended but made no further progress writing the last chapter of the Elements where applications of this operator was to have been presented, concludes that Hamilton therefore made no further progress in developing this operator, and so nothing was in fact lost. Both Tait and Joly point out that Hamilton kept his thoughts to himself until they were perfected to such a degree that he felt they could be made public, and both men remark that Hamilton rarely even wrote down his thoughts before they were perfected, much less publish them. So, it is an open question how far Hamilton went in his own mind developing this operator, for clearly he did not tell Tait everything he was thinking, even though Tait was personally very close to Hamilton, and Tait and Hamilton corresponded frequently on the subject of Quaternions. As we shall argue here, Hamilton did not even publish all his reasons for chosing the horizontal wedge and rejecting the vertical wedge. He was simply content to give his audience a reasonable explanation for why he had changed the symbol he initially used in the Southampton Paper to the one he found afterwards more appealing. His simple explanation "to avoid confusion" was really a subterfuge. As we shall see, there were more compelling reasons to pick the horizontal wedge, reasons Hamilton surely considered, but was not prepared to declare at that early time; this being consistent with his general policy of keeping his ideas secret until they were in the final perfect form for an audience. But, he did feel the pressure to tell his readers something, in 1846, because just after preparing the Southampton Paper, which Sir John Herschel saw and read, Hamilton saw the reason for turning this vertical wedge symbol about 90 degrees, but was not prepared to tell Herschel, or anyone else at that time, his about his metaphysically inspired guess that really prompted the change.

. 10. Hamilton, W. R., "Lectures on Quaternions", 1853, We reprint the entire Art.620 in History note 1853. See also footnote 2 for a discussion of a possible error in Joly's citations here.

. 11. Hamilton, W. R., "Elements of Quaternions", 2nd ed., Vol.I 1899, Edited by Charles Jasper Joly. See History note 1899. [See Elements2ed-I, pg.ix, preface to the second edition of Hamilton's Elements on Quaternions, by, C.J.Joly Dec.1898]

. 12. Joly, C. J., "A Manual of Quaternions", 1905. See History note 1905.

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. 13. pg.291. Phil.Mag., article "On Quaternions; or on a New System of Imaginaries in Algebra," by Sir William Rowan Hamilton, LL.D., V.P.R.I.A., F.R.A.S., Corresponding Member of the Institute of France, &c., Andrews' Professor of in the , and Royal Astronomer of Ireland., pp.278-293.

. 14. pp.530-531. Graves Life Vol.II.

. 15. Tait, P.G., "Scientific Papers", Vol.I, 1898, edited by P. G. Tait. See History note 1898.

. 16. Tait goes on to make a further note that from Graves Life III, p.94, it seems Hamilton never completed his ideas and wrote nothing, so that his fears of these notes being lost now seemed unfounded.

. 17. Reprinted in the Scientific Papers with "del" changed also. See the Edinburgh Proceedings and compare the reprint.

. 18. The idea of "Polarity" is one of the seven principles of occult law. Although some minor variations in the seven are found in different schools, "polarity" is always one of the seven. For example, the Rosicrucians give the seven Cosmic principles: (1) The Principle of Correspondence, (2) The Principle of Law and Order, (3) The Principle of , (4) The Principle of Rhythm, (5) The Principle of Cycles, (6) The Principle of Polarity, (7) The Principle of Sex. While, the Kybalion gives the seven mystical laws: (1) The Law of Mentalism, (2) The Law of Correspondence, (3) The Law of Vibration, (4) The Law of Polarity, (5) The Law of Rhythm, (6) The Law of Cause and Effect, (7) The Law of Gender. Metaphysics and The Occult often share the same ideas with a difference only on emphasis. Occultists tend to add an aura of mystery and secrecy around their ideas saying very little beyond the simple statements of principles, while metaphysicians tend to introduce much technical jargon and philosophical dialog while introducing the same ideas being rather much more verbose in their presentation, both groups tending nevertheless to dramatize the simple principles by their different styles. And both groups make the subject matter almost inacessible to all but the most determined student to penetrate into the dephts of meaning. The occult by its terseness leaving so much unsaid that the statements draw a blank in those minds too sluggish to think for themselves. While the metaphysicians by their excessive words bury the same ideas again in a mountain of philosophical babble often discouraging all but the most dedicated and determined student to penetrate its depths to find and understand the same simple truths again. [ Incognito, Magus, "The Secret Doctrine of The Rosicrucians", Illustrated with The Secret Rosicrucian Symbols, Printed by Yoga Publication Society, P.O. Box 148, Graceland Station, Desplaines, Illinois 60016, Copyright, 1949 by Occult Press, Chicago, Illinois, 256 pages, ISBN 0-911662-30-8, ref.pp.220-256, & Feb 2000 reprint, ISBN 1585090913; and Three Initiates, "Kabylion: A Study of the Hermetic Philosophy of Ancient Egypt and Greece," Sept 1997 Kessinger Publishing, 226 pages, ISBN 0766100804, and December 1999 DeVorss & Company Edition, ISBN 0911662251, also free online original 1912 copy, Copyright 1940 by Yogi Publication Society. ]

. 19. See History note 1866

. 20. pp.488-489; Graves, Robert Perceval, "LIFE of SIR William Rowan Hamilton," Vol.II, Andrews Professor of Astronomy in the University of Dublin, And Royal Astronomer of Ireland, etc.; 1885, Dublin: Hodges, Figgis, & Co.; London: Longmans, Green & Co., 719 pages, ..UFT Call No. QA.29.H2G7.v.2

. 21. For a complete reprint of the relevant brief passage from Joly's text, see History note 1905.

. 22. See History note 1846, for more information on the choice of this symbol.

. 23. James Clerk Maxwell, "Treatise on Electricity and Magnetism", 1873.

. 24. A copy of this letter appears as item no.277. Letter to Peter Guthrie Tait, 11 December 1867, on pp.328-334, of P.M. Harman's "The Scientific Letters and Papers of James Clerk Maxwell. Volume II, 1862-1973", published in 1995.

. 25. A copy of Tait's 13 December 1867 response letter appears at the end of item no.277. Letter to Peter Guthrie Tait, 11 December 1867, on pp.333-334, of P.M. Harman's "The Scientific Letters and Papers of James Clerk Maxwell. Volume II, 1862-1973", published in 1995.

. 26. Lectures on Quaternions, 1853. See History citation [Hamilton1853].

. 27. I have not been able to track down Tait's writings that uses the v-wedge as early as the 1867 Maxwell letter. But, it is obvious from the "reprint" of Maxwell's letter in Harman's work that Tait must have used the v-wedge somewhere by that time. -- unless Harman himself modified this letter in "reprinting it" without alerting us to the fact, thus introducing the v-wedge along with the h- wedge, which seems, however, unlikely -- The Proceedings, however, show Tait using the h-wedge up until the year 1870, and only after that time changing over to the v-wedge. This is an apparent paradox, or rather, historical inconsistency, that I have yet to resolve.

.

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28. Tait, P. G., "An Elementary Treatise on Quaternions", 1867. I have not seen this 1st edition of the text.

. 29. Tait, P. G., "An Elementary Treatise on Quaternions", 1875. This 2nd edition I have seen, it contains the v-wedge, as does the 1890 3rd edition.

. 30. For a complete reprint of Art.620, of the Lectures, see History note 1853.

. 31. Tait, P.G., "Sketch of Thermodynamics", second edition, revised and extended, 1877, Edinburgh: David Douglas. 162 pages [UFT Gerstein Call no. Physics Thermodyn T.]; [Note: by Peter G. Tait, Formerly Fellow of St. Peter's College, Cambridge; Professor of Natural Philosophy in the University of Edinburgh.]. See Chapter I. HISTORICAL SKETCH OF THE DYNAMICAL THEORY OF HEAT, pp.1- 3.

. 32. Harman, Vol.I, pg.4-5, says "On entering the University of Edinburgh in the autum of 1847 he attended Sir William Hamilton's class in Logic, Forbes' in Natural Philosophy, and Philip Kelland's class in Mathematics"..."Maxwell also attended Hamilton's class on Metaphysics". See also item no.16, pp.110-113, "Exercise on the properties of Matter, for Sir William Hamilton's Metaphysics class, Edinburgh University, 1848-49." [Harman quotes from Campbell and Garnett, Life of Maxwell; 109-13, because the original is document is now lost]...and refer to Harman's index in Vol.I and Vol.II under "metaphysics." His Professor here is Sir William Hamilton the Philosopher, not to be confused with Sir William Rowan Hamilton the inventor of quaternions.

. 33. Harman, Vol.I, pg.411, doc. no.101, pp.410-411, "Letter to Richard Buckley Litchfield, 4 July 1856." from James Clerk Maxwell-- quote: "I find I get fonder of metaphysics and less of calculation continually and my metaphysics are fast settling into the rigid high style, that is about ten times as far above Whewell as Mill is below him or Comte or Macaulay below Mill using above and below conventionally like up & down in Bradshaw."

. 34. Harman, Vol.II, pg.335, doc. no.278, "Letter to Peter Guthrie Tait" 23 December 1867; Here Harman also comments in footnote (4) "Tait scathingly dismissed metaphysical arguments on 'what is heat'? and any 'metaphysical pretender to discovery of the laws of nature' in his Sketch of Thermodynamics: 1-2.", which attitude we have already pointed out above.

. 35. "Letter to Mark Pattison", 7 April 1868, from J. Clerk Maxwell, reprinted as doc no.286, pp.358-361, in P.M.Harman's The Scientific Letters and Papers of James Clerk Maxwell, Vol.II, 1995, Cambridge Univ. Press. ISBN 0-521-25626-7.

. 36. The review article "On Quaternions" from Nature (25 December 1873), was not signed by Maxwell. It appears in that publication as an unsigned essay. However, in a footnote to the reprint of this article Harman writes, "Published (unsigned) in Nature, 9 (1873): 137- 8. The review can however be confidently attributed to Maxwell's authorship: the Nature archives confirm this attribution." see doc no.485, pp.951-955, in P.M.Harman's The Scientific Letters and Papers of James Clerk Maxwell, Vol.II, 1995, Cambridge Univ. Press. ISBN 0-521-25626-7.

. 37. With regard to the "metaphysical aspects of the method", Harman writes in a footnote "(3) In the 'Preface' to his Lectures on Quaternions (Dublin, 1853): 1-64, esp. 1-3, Hamilton describes 'Algebra as the SCIENCE OF PURE TIME', referring to his paper 'Theory of conjugate functions, or algebraic couples; with a preliminary essay on algebra as the science of pure time', Transactions of the Royal Irish Academy, 17 (1837): 293-422, esp. 293-7. In his Lectures on Quaternions: 2-3n ne makes explicit reference to 'passages in Kant's Criticism of the Pure Reason, which appeared to justify the expectation that it should be possible to construct, a priori, a Science of Time, as well as a Science of Space'.", see pg. 952, footnote (3), Harman, P. M., The Scientific Letters and Papers of James Clerk Maxwell, Vol.II, 1995, Cambridge Univ. Press. ISBN 0-521-25626-7.

. 38. Harman, Vol.I, General Introduction, pg.XiX, "There remains a puzzle about the fate of the corpus of Maxwell papers which passed into Garnett's hands. In his 1903 letter to Larmor Garnett recollected that letters written to Maxwell were generally returned to their owners, the correcpondents to whom they were addressed; while Maxwell's own papers and notebooks were returned to his widow. Comparison of the 'Papers of James Clerk Maxwell' held by Cambridge University Library with documentions included in the Life of Maxwell reveals that numerous letters and papers, including Maxwell's letters to his father and wife, his essays for the Cambridge Apostles, and his father's papers, which were published in partial extract in the Life of Maxwell and which would presumably have been returned to his widow, are no longer extant as autograph documents."

from pg.XXI, "The history of Maxwell's essays and family papers extracted by Campbell in the Life of Maxwell, and of other related material which he did not choose to print, can only be conjectured. It is possible that these documents were separated from Maxwell's scientific papers on his wife's death in 1886. His books being bequeathed to the Cavendish Laboratory while his other books were donated to the Cambridge Free Library, the city's public library. Garnett recollected that he had the impression that Maxwell's papers may have been transferred to Glenlair; and it is possible that the nonscientific and family papers were handed over to the custody of Maxwell's heir. If so, the papers were no doubt destroyed when Glenlair was later destroyed by a fire which reduced the building to a shell."

. 39. Harman Vol.II, 1995, pg.29.

. 40. Harman Vol.II, 1995, pg.30, footnote (90).

.

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41. Treatise on Electricity and Magnetism, 2nd ed. (I have not seen the 1st ed. of 1873). The editor of the 2nd edition claims Maxwell completely rewrites the first nine chapters, for this new edition, leaving the intended revision of the remainder unfinished upon his death. The remainder of the 2nd ed. is therefore reportedly identical with the 1st ed. While Price instructs Maxwell to use coordinate notation, there are indications from other letters and writings of Maxwell that he himself concurred with this view. Thus, the exact extent to which Price's directive was felt as a restriction by Maxwell is not that clear. Nevertheless, what is clear is that Maxwell in fact exercised restraint in using quaternions in his "published" works.

. 42. Harman Vol.II, pg.954. see also footnote 36.

. 43. Campbell, "Life of Maxwell".

. 44. Harman Vol.I, 1993, General Introduction, pg.xx.

. 45. We should emphasise the point here that the review article "On Quaternions" was not signed by Maxwell. However, as mentioned in note 36, Harman assures us that it is Maxwell's work. We can only conclude that Maxwell did not feel confident enough in his position as a Mason to reveal his identity to the public, but then why did he write the revealing comments at all? Certainly, he must have joined the order sufficiently long ago to have dared to write anything remotely revealing, considering that the solemn nature of the oaths of secrecy would require some time to digest and their seriousness perhaps worn a bit thin for a new recruit to be that bold to think to judge for himself what can be said and what cannot, and yet not so long ago that the novelty of the ideas he was being introduced to had yet worn off, so that in a burst of youthful enthusiasm he could not hold himself back from revealing the content of those relatively new thoughts that were then inspiring him, and yet again not feeling senior enough to either sign his name to his own works or choose the path to keep silence; not to forget that this action necessarily implicates the cooperation and understanding of the publisher.

. 46. According to a few online websites, the Royal Society of London was founded in 1660 A.D. by men who were Masons, and therefore it is not unreasonable to expect that many men of science would have been members of a Masonic Order. Sir W. R. Hamilton, and J. C. Maxwell, could very well have both been initiated into that Secretive Order. Note, however, that other sites claim that the official Freemasonry, as it is known today, did not exist until the year 1717, when the order became somewhat public. See Freemasonry and the Royal Society., and some History; etc..

. 47. Harman, Vol.II, pg.953-4, "There are thus three geometrical quantities having direction, and the more than magical power of the method of Quaternions resides in the spell by which these three orders of quantities are brought under the sway of the same system of operators.

The secret of this spell is twofold, and is symbolised by the vine-tendril and the mason's rule and square. The tendril of the vine teaches us the relation which must be maintained between the positive direction of translation along a line and the positive rotation about that line. When we have not a vine-tendril to guide us, a corkscrew will do as well, or we may use a hop-tendril, provided we look at it not directly, but by reflexion in a mirror.

The mason's rule teaches us that symbol, as written on paper, is not a real line, but a mere injunction, commanding us to measure out in a certain direction a vector of a length so many times that of the rule. Without the rule the symbol would have no definite meaning. Thus the rule is the unit of the Quaternion system, while the square reminds us that the right angle is the unit versor.

The doctrine of the unit is a necessary part of every exact science, but in Quaternions the application of the same operators to , vectors, and areas is utterly unintelligible without a clear understanding of the function of the unit in the science of measurement.

Whether, however, it is better to insinuate the true doctrine into the mind of the student by a graduated series of exercises, or to inculcate it upon him at once by dogmatic statements, is a question which can only be determined by the experience of a new generation, who shall have been born with the extraspatial unit ever present to their consciousness, and whose thoughts, quided by the vine-tendril along the Quaternion path, shall turn always to the right hand, and never to the left.

...Kelland - as he has elsewhere told us - finds but little difficulty in teaching the elements of the doctrine of Vectors to his junior classes, Hamilton himself, the great master of the spell, when addressing mathematicians of established reputation, found, for his Quaternions, but few to praise and fewer still to love."

One wonders, when examining Maxwell's particular choice of words and analogies here in these phrases whether he is not trying to preach to two completely different audiences simultaneously ! What is all this business about students being born to have "thoughts" that "shall turn always to the right hand, and never to the left" ? Is Maxwell still talking about the same Quaternions from Hamilton here? Is he hinting at something else? Certainly, his choice of words shows his own thoughts intermingling different areas of interest. Mathematicians and scientists, don't usually describe their art as some kind of "spell" cast by a magician.

. 48. Wilson says, "Kelvin summarised Stoke's areas of expertise: 'Hydrodynamics, elasticity of solids and fluids, wave-motion in elastic solids and fluids, were all exhaustively treated by his powerful and unerring mathematics.", see pg.43, of Wilson, David B., "Kelvin and Stokes: A Comparative Study in Victorian Physics", 1987, Adam Hilger, Bristol, 253 pages, ISBN 0-85274-526-5, UFT Gerstein Lib. Call no. QC.16.K4W55.1987.c.1.PASC; Stokes taught for many years at Cambridge University, being the favorite Professor for those seeking to take the Mathematical Tripos, and he was Maxwell's professor in 1854, and Tait's in 1851. [See Wilson, pg.52.. "Stokes students included P. G. Tait (1851), Maxwell (1854), ..., W. K. Clifford (1865), J Larmor (1879) who was Stokes successor as Lucasian Professor of Mathematics at Cambridge University in 1879"..], Stokes and Thomson often disagreed on physical interpretations, but Stokes "mathematics" was the foundation of Maxwell's Theory of Electricity and Magnetism, although it was Thomson's early papers on Electricity and Magnetism that inspired Maxwell's own development of the Theory. Ironically, Thomson afterwards disagreed with many of the changes Maxwell makes. Thomson's copy of Maxwell's 1873 Treatise contains many handwritten comments concerning his objections to Maxwell's Theory, a complete study of these notes have yet to appear in print, as Wilson writes, "Kundsen has gone some way towards that study in recent publications. He identified points in Maxwell's theory that Thomson regarded as wrong or deficient: its concept of transverse electrical waves, its idea of displacement currents producing magnetic effects, its imperfect

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dynamical foundation, and its lack of an explanation of interactions between ether and matter." [see Wilson, pg.9]. Maxwell based his Theory of Electricity and Magnetism on Hydrodynamics, the mathematical training of which he received at the expert hands of Stokes, while his physical intiution came from Faraday and Thomson.

. 49. Letters to Stokes and Thomson should have been returned to them, acording to Garnett, but it seems exactly what transpired in this whole area is a bit clouded. See Harman Vol.I, General Introduction.

. 50. From the Maxwell writings that have survived and are published, I have seen no hints that Maxwell thought of linking thermoelectricity and electromagnetism, nor any hints as to what he might have thought privately about quaternions that he did not already make explicit and obvious in his publications. Except that, from his oblique reference that men "shall turn always to the right hand, and never to the left" in the review article "On Quaternions" one could conjecture that Maxwell himself felt that only to "right thinking men" should certain things be revealed. But, that again is probably a bit of a stretch.

. 50a. Knott, Cargill Gilson, "Life and scientific work of Peter Guthrie Tait, supplementing the two volumes of Scientific papers published in 1898 and 1900," published 1911 by Cambridge University Press, London. pp.244-245, reprints Maxwell's last letter to Tait, of 28 August 1879. The University of Cornell, New York, has digitized the entire work and made it freely available online. The relevant web pages are located here, Cornell Index Page. The digitized page numbers are offset from the actual books page numbers, so to get the true book page 245 click page245, or enter 251 from the index page given; and for the prior true book page 244 click page244, or enter 250 from the index page given.

. 50b. McAulay, Alexander, "Ulility of Quaternions in Physics," 1893, Macmillian and Co., New york. McAulay writes about his discussions with Tait on the innovations to quaternion differentiation and Tait's objection to them in the preface of this work. The University of Cornell has digitized this entire text and made it freely avaliable online here, Cornell Index Page. The digitized page numbers are the same as the true page number of the book, (unlike some other Cornell digitization books), and the relevant pages are ix, x, xi, and these pages number may be entered from the index page also. The discussion, while reported by McAulay here in 1893, had apparently been ongoing from about 1884, and some details were included by Tait in his 1890, 3rd ed. of the "Elementary Treatise on Quaternions."

. 50c. Crowe, Michael J., "A History of Vector Analysis. The Evolution of the Idea of a Vectorial System," 1993, (1985, 1967), Dover, ISBN 0-486-67910-1, pg.259, etc. Crowe discusses the battle between vectors and quaternions in the text, and gives the last date for the publication of the Bulletin as 1913.

. 51. Hamilton, W. R., "Elements of Quaternions", 2nd ed., Vol.I, 1899, Vol.II, 1901, edited by C. J. Joly. See also History notes 1899 and 1901.

. 52. Joly, C. J., "A Manual of Quaternions", 1905. See also History note 1905.

. 53. Art 57, pp.74-77, "A Manual of Quaternions", by C.J.Joly, 1905. For a reprint of relevant parts of this article see History note 1905.

. 54. Page 76, Exercise 11, Art 57, "A Manual of Quaternions", by C.J.Joly, 1905. See also History note 1905.

. 55. Although Heaviside cast aside Quaternions and wrote about his own version of vectors, he did, nevertheless, encounter the same left-right question when dealing with vector products or cross products. Joly certainly would have read Heaviside's view that the left- right issue was a matter of "convention." See History note 1891.

. 56 pp.135-143, Vol.III, Appendix K; Heaviside, Oliver, 1850-1925, Electromagnetic Theory, Vol I, 1893, 466 pages, "The Electrician" Printing and Publishing Company; Vol II, 1899, 547 pages one printing, 542 pages another printing !! (check difference); Vol III, 519 pages, 1925, New York, D. Van Nostrand (first published 1912, repr. 1922, 2nd repr. 1925), [ see also Faq--Heaviside Numbers ]

. 57 For a discussion on Heaviside's algebra system see Faq--Heaviside Numbers.

. 58 pg.430; Heaviside, Oliver, "On the forces, stresses and fluxes of energy in the electromagnetic field," read before the Royal Society in June 1891, ; Royal Society of London. Philosophical Transactions. Series A, 183, (1892), pp.423-480. Received June 9, Read June 18, 1891. Delayed for publication until 1892 because of late footnotes added May 11, 1892. [ See also History note 1891 ]

. 59 Pertti Lounesto is the first person I've seen use the backslash '\' for the other way to divide in quaternions, and I've since adopted this notation. His remarks in sci.math usenet postings on quaternions represented these left and right divisions in the forms, q\p and p/q, for the products, q^-1 . p and p . q^-1, respectively.

. 60 Letter From Sir W.R. Hamilton to his father Rev. James Hamilton, Observatory, September 11, 1846. (See also, Sir W.R. Hamilton..his treatise on Algebra as the Science of Pure Time, in the Transactions of the Royal Irish Academy, vol.xvii,1837.) [See

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Graves-LifeIII, vol.iii, pp.633-634, pg.635-footnote] And See also discussion in History note 1998.

. 61 JD's comments in sci.math, sci.physics, usenet groups on "Hamilton Visualization" can be viewed with in the usenet archieves with www.deja.com using these keywords, and the acronym, [HV].

. 62 pg.136, v.3, Heaviside, Oliver, "Electromagnetic Theory", EMT. See also Faq--Heaviside Numbers.

. 62a Pertti Lounesto pointed out this particular result, q = (b/a)1/2, to me, for the solution to the quaternion that rotates the initial into final vectors, when I posted the related problem on the sci.math usenet . For more complete discussion of the rotation problem, see this website's article 2001-08-04 . Quaternion Unit: Rotation and Inversion in the Great Pyramid, and especially it's footnote 7.

. 63 Correlation.

. 64 Hume, "An Enquiry Concerning Human Understanding," Hume Online.

. 65 In the paper "Physical Space as a Quaternion Structure: Maxwell's Equations", we demonstrate how the quaternion, with both scalar and vector parts united, naturally leads us to a form of Maxwell's Equations which includes addtional terms that can link thermoelectricity to electromagnetism. See hypcx-p20001015.html.

. 66 See hypcx-p20001015.html.

. 67. The best extension of Hamilton's operator from 3 to 4 dimensions is by no means immediately obvious. While it seems that "time" is the natural extra degree of freedom that should play a role in the 4th , the exact form of time's involvement is somewhat non-intuitive. We have taken our clues from relativity, which treats the quadruple (ct,x,y,z) as a four-vector, to suggest the 4-d operator (with c=1),

d/dr = @/@t + i@/@x + j@/@y + k@/@z,

as the appropriate extension to Hamilton's , and we use this form exclusively throughout our introductory presentation. But, quaternions, and other indications from physical science, suggest that there are conditions where a linear time measure is really on the same level as the square of a spatial length measure. With the usual dimensional analysis symbols, T, L, M, for "time," "length," and "mass," this relationship can be briefly expressed as T = L^2. We see this relationship on the large scale in the motion of planets, where Kepler's Law says planets sweep out equal areas in equal units of time, which suggests L x L = T. And we see this again on the small scale where the Principle of Complementarity and the Uncertainty Relation tell us that x.p = h, which we can write [x.m.x/t] = [h], indicating that [x^2] / [t] = [h/m], again hinting that L^2 is proportional to T. And again, the origins of the 3-d space (i,j,k) we know so well can be seen to come out of a 1-d time measure via whatever physical process could claim representation by the mathematical operation of the square root. Indeed, the genesis of quaternions contains the suggestion of this correspondence, t = x^2, as can be conjectured from the basis, -1 = i^2, etc. Length is proportional to the square root of time, not to time itself. And therefore a more appropriate extension might take the alternative form,

d/dr = @1/2/@t1/2 + i@/@x + j@/@y + k@/@z,

where the power (1/2) in the time-differential represents the fractional differentiation of order 1/2, otherwise known as the semi- derivative. Oliver Heaviside has actually explored the application of fractional calculus to Electromagnetic Theory, and has used this same time-semi-derivative to re-write equations that would otherwise appear as second order, d2F/dx2 = b.dF/dt, to be first order in the space variables instead, dF/dx = b'.d1/2F/dt1/2, and then applied his operational methods and those of fractional calculus to find the solution.

He demonstrates that the fractional operators, d+1/2/dt+1/2 and d-1/2/dt-1/2, are equivalent to the conductance operator and resistance operator, respectively, of a transmission cable; the former transforming potential into current, while the latter turning current into voltage. And with this, he is the first to find a physically meaningfull interpretation of fractional differential operators. Perhaps the most significant property of the semi-derivative with respect to time is that it uniquely defines the instant of creation of a signal, because events depending on the square-root of a time variable cannot then be arbitrarily shifted about by linear translations along the time-axis. This may prove to be a benefit in the description of transient electromagnetic phenomena, justifying the use of the fractional calculus approach to modeling some things in electricity. However, rather than complicate the theory early with issues of fractional calculus, when we already have a fair level of complexity to deal with from quaternion calculus alone, we've elected to explore the results obtained in electric theory without the benefit of the fractional calculus. Whether the more productive extension to Hamilton's operator turns out to be,

d/dr = @/@t + , ...... -or- ...... d/dr = @1/2/@t1/2 + ,

or perhaps some combination of these two, remains to be seen. Only further explorations with these two operator extensions can suggest which is the best. For the moment, we ignore the alternative forms, but simply wish to alert the reader to the reality that our approach is not the only one that can be taken.

______Date document last modified yyyy-mm-dd: 2002-05-07 (original date rev .01, 2001-07-19)

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