# Quaternions to Vector Analysis

by Kathy Joseph

I just finished a video on a history of the quaternions and a biography of their inventor, William Rowan Hamilton. In it, I stated, I hope pretty convincingly, that the basic features of vector algebra ALL came from Hamilton: the scalar, the vector, the dot product (or, at least the negative of the dot product), the cross product, the del function (also called the Nabla function), the divergence (or at least the negative of the divergence) and the curl all were created if not named by Hamilton between 1846-1847 specifically to help with describing physics systems. Although quaternions are quite popular with mathematicians and computer programmers, the majority, and I mean the vast majority of physicists only know of Hamilton for the Hamiltonian and have NO idea that he also made the preponderance of the mathematics that we use in advanced physics.

But why? Well, it is intertwined with the history of Maxwell’s equations and the life and personality of a man named Peter Tait. So, to understand, I would like to start in 1846 with a friendship between two teens named Peter Tait and James Clerk Maxwell. Ready for a long and surprisingly emotional video? Let’s go!

Part 1: Peter Tait & James Clerk Maxwell (1846-1856)

When Peter Tait met James Clerk Maxwell at a prestigious high school called Edinburgh Academy in late 1846, 15-year-old Tait agreed with his classmates that Maxwell was an odd duck. Tait even joined his classmates in calling Maxwell daft or daftie (meaning crazy).[1] Tait said that the big problem was that he and many of his friends were “quite innocent of mathematics” and Maxwell’s “absorption in such pursuits” caused the other students to avoid Maxwell and he made no friends. However, Maxwell was championed by a teacher named Dr. James Forbes and after Maxwell “surprised his companions” by winning awards, Tait decided to start talking to Maxwell. Soon they became best buds and Tait recalled fondly how, “From this time forward I became very intimate with him, and we discussed together, with schoolboy enthusiasm, numerous curious problems.”[2]

Although Tait already had an interest in science and mathematics before he met Maxwell, it was arguably this friendship (and the influence of Dr. Forbes) that transformed Tait into a mathematical physicist. Inspired, before he even turned 18, Tait left the Academy in 1848 and moved to Cambridge to focus on becoming a mathematician and mathematical scientist.[3] Maxwell followed him two years later in 1850.[4]

Tait was a dedicated student (winning first or first wrangler in something called the Tripos in 1852) but he found his work to get there to be painful. Scrawling his frustration about studying for this test that it was: “purgatory,” “torment” and at one time in all capitals (and in Latin) “L’ENFER!” or “HELL!”[5]

Maxwell also didn’t have a great time at Cambridge, writing to his friend Lewis Campbell “Facts are very scarce here” and that his teachers “grind up these subjects.” Intellectually, Maxwell complained, the Cambridge student “starves while being crammed. He wants man’s meat, not college pudding. Is truth nowhere but in mathematics?”[6] Still, Maxwell was too engrossed in his mathematics to give up, even though he felt that most of the students were just as disdainful of him at Cambridge as they were at the Academy. Maxwell came in second in Mathematical Tripos in January, 1854 and was finally free to study what he wanted to, which Maxwell referred to as “enter[ing] the unholy estate of bachelorhood.”[7]

It was for that reason that in February, 1854, that 22-year-old Maxwell wrote to his friend and mentor William Thomson asking for his advice on how to “attack electricity.”[8] Specifically, Maxwell wanted to know if he should begin with Faraday who used no mathematics or with the people that Faraday seemed to be in conflict with like Ampère who were math based. Thomson then told Maxwell that Faraday was not at all in conflict with the other scientists and he should start with Faraday. When Maxwell read Faraday’s Experimental Researches he not only agreed with Thomson but also felt that, “I perceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols.”

So, Maxwell just decided to try to convert Faraday’s ideas from Faraday’s “math” into “the ordinary mathematical forms.[9] By December 1855 and February 1856, Maxwell published a series of papers “On Faraday’s Lines of Force” where he modeled Faraday’s electric and magnetic lines of force as theoretical tubes of incomprehensible fluids (so he could use fluid mechanics) with charges either being sources or sinks of these “fluids.”[10] Among the many equations that Maxwell derived in these papers was that there were areas of equal “pressure” p, which were equivalent of both the voltage from a battery or what they called the “electric tension” and the potential in static electricity.[11]

In addition, inspired by how Thomson modeled pressures and temperatures, Maxwell decided that, “it is easy to see that these surfaces of equal pressure must be perpendicular to the lines of fluid motion.”[12] Mathematically, he realized that these “lines of fluid motion” were a vector with values in the x, y and z direction which he designated \alpha, \beta and \gamma , where:

\alpha=\frac{dp}{dx}\ \ \ \ \ \ \ \beta=\frac{dp}{dy}\ \ \ \ \ \ \ \gamma=\frac{dp}{dz}[13]

Those of you who are familiar with vector physics, can clearly see that this is equivalent to the modern equation for the relationship between the electric field and the electric potential in the static system i.e.:

\vec{E}\ =-\nabla\phi

If \phi is used instead of p, and the electric field, \vec{E} is defined as -(\alpha, \beta, \gamma).

Now Maxwell was an excellent mathematician, but he didn’t realize that this special triangle operator that is needed for the gradient had already been “invented” 8 years earlier in 1847 by the Irish mathematical scientist William Rowan Hamilton as part of his conceptually tricky quaternions.[14] Instead, it was Tait, not Maxwell who first appreciated quaternions and that brings me to.

Part 2: Tait, Hamilton and the Quaternion (1854-1867)

In September of 1854, just after Maxwell was communicating with Thomson about “attacking electricity,” Tait was offered a position as a professor of mathematics at Queen’s college.[15] It was at this new job that Tait started to discuss with a coworker about his struggles with quaternions which led to a friendship with the inventor of quaternions, William Rowan Hamilton.[16]

Hamilton created quaternions 11 years earlier as an extension of complex numbers. So instead of having a complex number (x + iy) where i is the imaginary number \sqrt{-1} where  quaternions have what you might call a very complex number, (w + ix + jy + kz) with a real number and three imaginary numbers, i, j and k, which is why in quaternions i^2=j^2=k^2=-1

In addition in 1846, Hamilton defined the real term the “scalar” and terms withi, j or k the “vector.”[17] In this way, Hamilton could use his quaternion multiplication rules to multiply vectors which would give two results: a scalar that was equivalent to the negative of the modern dot product, and a vector that was equivalent to the modern cross product.

Even more intriguing to Tait, in 1847, Hamilton created a new operator (that I am going to call the del operator, for reasons I will explain later), which he initially symbolized with an upside-down delta or triangle, but then changed to a sideways triangle. Anyway, this unnamed triangle was defined as[18]:

\mathrm{\nabla}=\frac{d}{dx}i+\frac{d}{dy}j+\frac{d}{dz}k

The reason he created that operator is because in physics problems the term {[\left(d/dx\right)}^2+\left(d/dy\right)^2+\left(d/dz\right)^2] came up regularly and, in Hamilton’s quaternion mathematics \nabla has no vector term and the scalar result is the negative of this function, ie:

\nabla^2=-{[\left(d/dx\right)}^2+\left(d/dy\right)^2+\left(d/dz\right)^2]

Interestingly, Hamilton also noted that if you took the del of a vector Q with x, y and z of (t, u, v) then you get two results,[19] a scalar (which is equivalent to the modern -\nabla \bullet Q ):

Scalar \nabla Q =-(dt/dx+du/dy+dv/dz)

And a vector result (which is equivalent to the modern \nabla \times Q) or:

Vector \nabla Q = (dv/dy-du/dz)i+(dt/dz-dv/dx)j+(du/dx-dt/dy)k

However, Hamilton did not, as far as I can tell, make the simple leap to determine what happened if you took the del of a scalar, p, which would, according to his rules result in a simple vector (which is equivalent to the gradient of a function), ie:

\mathrm{\nabla p}=\frac{dp}{dx}i+\frac{dp}{dy}j+\frac{dp}{dz}k

Which, clearly, would be interesting to anyone dealing with electric or temperature potentials.

Hamilton then combined many of his articles about quaternions in a book titled, unimaginatively, “Lectures in Quaternions,” written in 1853,[20] which is how Tait initially learned of quaternions. As Tait told Hamilton five years later, “I attacked your volume on Quaternions immediately on its appearance.”[21] However, when it came to this triangle or del function, Tait had trouble. In August of 1858, Tait wrote to  Hamilton that “since I left Cambridge… [I] have been busy at the Theories of Heat, Electricity, etc. Your remarkable formula for \frac{d^2}{{dx}^2}+\frac{d^2}{{dy}^2}+\frac{d^2}{{dz}^2} as the square of a vector form… appear to me to offer the very instrument I seek, for some general investigations in Potentials.”[22]

Soon the letters were flying back and forth between Oxford, England, where Tait was working and outside of Dublin, Ireland where Hamilton lived and worked as the royal astronomer. By April 1859, Hamilton wrote Tait, “Do you not feel, as well as think, that we are on a right track, and shall be thanked hereafter? … [When I created quaternions, a friend] wrote to me, nearly as follows: – ‘I suspect, Hamilton, that you have caught the right sow by the ear!’ Between us, dear Mr. Tait, I think that we shall begin the SHEARING of it.”[23]

However, Hamilton was busy writing his second book “Elements of Quaternions,” which he was sure would complete very soon telling Tait that “The only thing I ask was that you would not publish a separate work before the appearance of the Elements. I shall be charmed, for both our sakes, to set you free as soon as possible.”[24]

Meanwhile, Tait was diverted from quaternions by Maxwell’s mentor William Thomson. See, in 1860, Tait and Maxwell’s old teacher Forbes who had moved to Edinburg University had retired, and Tait beat out Maxwell for the position (mostly because Maxwell was known as being a terrible instructor).[25] Tait’s new job pleased Thomson who had been worried that “a mere nobody” “has a good chance for the chair.”[26] Thomson then decided to collaborate with Tait on a series of Physics books, writing his brother, in January of 1862: “I have been projecting a book on Natural Philosophy (elementary and non-mathematical) along with [Professor Tait] and as he has very great executive energy and facility in writing, I hope we may soon get a volume 1 out.”[27]

However, both Tait and Thomson’s “elementary” physics book and Hamilton’s quaternion book turned out to be more challenging than they were expecting, and the years passed with no books published. Tait and Thomson were young healthy men in their late 20s and early to mid-30s, but Hamilton was in ill health, in his late 50s and had trouble with gout and possibly overwork (but not drunkenness as I discussed in my last video). In addition, despite Hamilton’s and Tait’s enthusiasm in quaternions, they seemed to have trouble getting anyone else interested, perhaps because the math was in 4-dimensional imaginary space. In 1865, Maxwell wrote Tait, “Does anyone write quaternions but Sir W. Hamilton and you?”[28]

Then, tragically, in late August of 1865, when Hamilton had a terrible bronchial infection, Hamilton realized that he wasn’t going to live much longer and reached out to Tait. According to Tait: “Sir W Hamilton, a few days before his death, urged me to prepare my work as soon as possible, his being almost ready for publication. He then expressed… his profound conviction of the importance of Quaternions to the progress of physical science; and his desire that a really elementary treatise on the subject should soon be published.”[29]

Poor Tait was torn, he was being pressured by Thomson to finish his physics book, but he felt he needed to honor Hamilton’s deathbed request. Perhaps because of this, Thomson began to loathe quaternions, a feeling that Thomson felt for the remainder of his life. Thomson even recalled that he basically taunted Tait multiple times that they could include quaternions into their book “if he could only show that in any case our work would be helped by their use.”[30] But, as this was supposed to be an elementary and non-mathematical textbook, of course that did not happen (although the book ended up being far less elementary and non-mathematical than Thomson had originally planned.)

Years later, in Tait’s obituary Thomson recalled that this “war” was part of their friendship and that even though he and Tait often “had keen differences … on every conceivable subject – [including] quaternions… We never agreed to differ, always fought it out. But it was almost as great a pleasure to fight with Tait as to agree with him.” [31] Anyway, back in 1867, Thomson and Tait published their first (and it turned out their only) book on Natural Philosophy. As Thomson would sign his letters with a simple T, Tait began to sign his T’, Maxwell referred to their book as, “T and T’” and the name stuck.[32]

That same year as the T and T’ book, 1867, Tait finished his first book on Quaternions. In this book, Tait expanded on the uses of quaternions and of Hamilton’s triangle operator which he, for some reason, rotated back to be in its current orientation.[33] Tait felt that due to his work with William Thomson “in the laborious work of preparing a large Treatise on Natural Philosophy,” his quaternion book was incomplete, writing, “I regret that I have so imperfectly fulfilled this last request of my reverend friend.”[34]

Despite this, or perhaps because of his plea to others to contribute just after Hamilton’s untimely death and Tait’s simpler style, quaternions slowly started to be more discussed in mathematical circles. Which brings us back to James Clerk Maxwell and his equations.

Part 3: Maxwell, His Equations, and Quaternions (1856-1879)

Back in February, 1856, when Maxwell was a student finishing his first set of papers on lines of force, he took a job as a professor Marishal College in Aberdeen, Scotland and grew a beard to make himself look professorial. He then wrote to his friend Campbell, “no one here seems to think me odd or daft. Some did at Cambridge, but here I have escaped.”[35]

Marishal College is also where Maxwell met the love of his life, Katherine Dewer and in June, 1858 they married, a relationship that his friend Campbell called “one of unexampled devotion.”[36] Maxwell’s wife had no background in science or math, but she was interested and soon started contributing, where, according to Maxwell, his “better half” contributed to experiments in heat transfer. Then, in 1860, Maxwell’s college was combined with another to form the University of Aberdeen and Maxwell found himself without a job. However, Maxwell quickly found a new job at King’s College, London after Tait left the position open.

It was at King’s College at the end of 1861 and the beginning of 1862 that Maxwell first wrote “Maxwell’s equations.” What happened is that Maxwell heard about a model of the atom as a charged nucleus with an electric atmosphere from a fellow Scotsman named William Rankine. With this more realistic “molecular vortexes” model, Maxwell wrote the 4 “modern” Maxwell’s equations, albeit with different units, and without vectors in a sea of 165 equations.[37]

Additionally, with his laws, Maxwell discovered that in a vacuum, a vibrating electric field would create a vibrating perpendicular magnetic field and the reverse moving at a speed that depended on the square root of an electric constant  to a magnetic constant . This was particularly inspiring to Maxwell as, in 1857 a German team named Dr. Weber and Kohlrausch experimentally measured this ratio and found that:

\sqrt{\frac{k}{\mu_0}}=3.1\ x\ {10}^8\ m/s[38]

In February, 1862, Maxwell stated outright: “The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of Mr. Kahlrausch and Weber, agrees so exactly with the velocity of light… that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.[39]

These ideas were so exciting that in 1864, Maxwell wrote another paper expounding on what these ideas meant about light and about “the Electromagnetic Field” (a term Maxwell created for this paper). (Note that Faraday had suggested that maybe light was a vibration of electric and/or magnetic lines of force 18 years earlier in 1846, which is why Maxwell wrote that “The electromagnetic theory of light, as proposed by [Faraday] is the same in substance as that which I have begun to develop in this paper, except that in 1846 there were no data to calculate the velocity of propagation.”[40])

In April 1865, Maxwell decided that he was so busy with his theories and experiments which he “could not undertake as long as I had public duties” and so he retired from teaching and he and his wife Katherine moved back to Scotland.[41]

After Thomson and Tait published their (T and T’) Treatise on Natural Philosophy in 1867, Maxwell began contemplating writing his own “elementary” textbook on electricity and magnetism (or what Tait called a “Senate-House Treatise.”) Tait was thrilled, writing Maxwell, “I am delighted to hear you are going to do a Senate-House Treatise on Electricity. The sooner the better!”[42]

Soon Maxwell was communicating endlessly with both Tait and Thomson and debating the theory, the mathematics and the best method of communication of these complicated ideas. After multiple distractions, in November 1870, Maxwell started to seriously think about using quaternions. Maxwell then wrote Tait with a series of proposed names for quaternion operators as, “I am unlearned in quaternion idioms…[and] I want phrases of this kind to make statements in electromagnetisms and I do not wish to expose either myself to the contempt of the initiated, or Quaternions to the scorn of the profane.”[43]

For example, Maxwell named the triangle (\nabla) operator the “alted” function (for delta backwards) and the scalar part of the alted times a vector (or -\nabla \bullet \textbf{A}) to be the “convergence”.

Maxwell then went through several names for the vector part of the alted times a vector (or \nabla \times \textbf{A}). What did Tait think of “twist” “turn” or “version”? Maybe “Twirl” is “sufficiently racy,” or perhaps a “pure mathematician” might prefer “curl (after the fashion of Scroll)”?[44]  By October, 1872, Maxwell wrote to his friend Campbell that “I am getting converted to Quaternions, and have put some in my book.” He also added that Hamilton’s function, \nabla “occurs continually.”

Amusingly, in these letters Maxwell started to informally call the triangle function Nabla after a type of harp as a joke so that he could quip that he was a “Nablody” for using it. [45] He also liked to ask Tait if he was “still harping on that Nabla?”[46]

Maxwell finished his book in 1873, where he called the triangle operator the alted operator, the scalar of alted times a function the “convergence” and the vector of the alted times a function the “curl” and the del squared as “Laplace’s Operator,” all names we use to this day.[47]

Then, tragically, in the spring of 1877, James Clerk Maxwell started having severe health issues, and by October, 1879, 48-year-old Maxwell was told that, like his late mother, he had terminal stomach cancer and only had a month to live.[48] Maxwell took the news stoically, and was mostly concerned with his wife who had chronically delicate health. By November 5th, he passed away. His cousin and good friend Colin Mackenzie was there at the end and I think it is worth repeating his words in full: “Maxwell was being held up in bed, struggling for breath, when he said slowly and distinctly, ‘God help me! God help me wife!’ he then turned to me and said, ‘Colin you are strong, lift me up;’ He next said, ‘Lay me down lower, for I am very low myself, and it suits me to lie low.’ After this he breathed deeply and slowly, and, with a long look at his wife, passed away.”[49] Tait and Thomson were among the many who were devastated. Tait cried “I cannot adequately express in words the extent of the loss which his early death has inflicted not merely on his personal friends, on the University of Cambridge, on the whole scientific world, but also, and most especially on the cause of common sense [and] of true science.”

However, Tait consulted himself, “Men of his stamp never live in vain; and in one sense at least they cannot die. The spirit of Clerk-Maxwell still lives with us in his imperishable writings, and will speak to the next generation by the lips of those who have caught inspiration from his teachings and example.”[50] Now Tait felt that he had two great works to promote after their early deaths: Maxwell’s and Hamilton’s. It is no surprise then that Tait became personally deeply invested, one might even say territorial, in having Maxwell’s equations being written in quaternions. In this he was going to be bitterly disappointed, which brings me to an American physicist named Josiah Williard Gibbs.

Part 4: Gibbs (1873-1884)

Josiah Williard Gibbs, who was 7 years younger than Maxwell and Tait, was a long-term collaborator on the subject of statistical mechanics (a term that Gibbs coined). So it should be no surprise that Gibbs read Maxwell’s 1873 book on electricity and magnetism. That is where Gibbs learned about quaternions and he decided that, “it was necessary to commence to mastering those methods.” However, he found that “respect to the operator \nabla as applied to a vector I saw that the vector part and the scalar part of the result represented important operations, but their union (generally to be separated afterwards) did not seem a valuable idea.”[51]

For that reason, Gibbs put together a new mathematical shorthand based on Hamilton’s quaternions with two two kinds of multiplication: a scalar multiplication and a vector one.  Then, between 1881 and 1884, Gibbs put together a little pamphlet about his new mathematical methods to help his students understand Maxwell’s equation. In this pamphlet, Gibbs called the scalar multiplication the “direct product” and noted it with a dot.[52] Gibbs also called the vector multiplication the “skew product” and noted it with a x or a cross.

I think that it is startling to compare Gibbs’ scalar product and vector product to Hamilton’s scalar and vector results of multiplying two vectors from his 1846 paper. You can clearly see how Gibbs’ “direct product” is identical to the scalar product without a negative sign and the “skew product” was identical to the vector result from a quaternion. Shoot, Gibbs even called his directions i, j, and k, after Hamilton’s notation but now i, j, and k are now unit vectors and not equal to the square root of negative one.

With this new mathematics, Gibbs needed a name for the negative convergence. Conveniently Gibbs had read a physics book by the mathematician Williard Clifford, who wrote in 1878 about this very problem where he noted that: “Prof. Clerk Maxwell calls the quantity -E the convergence of σ. We might perhaps therefore call E itself the divergence of σ“[53]

That is why in Gibbs’ pamphlet Gibbs related Clifford’s “divergence” and Maxwell’s “curl” to Hamilton’s del function when he wrote that “\mathrm{\nabla}\bullet w is called the divergence of w and \mathrm{\nabla}\times w its curl.[54]

In this case, you can see how Gibbs created the divergence from the negative of Maxwell’s convergence and the curl directly from Maxwell’s definitions in his 1873 book.

This is clearly, aside from the minor differences of the names of the products and the position of the dot in the dot product, modern vector algebra.

After Gibbs created this mathematics, he read another mathematician named Grassmann who had come up with a geometrical model in 1844 which actually had a separate scalar product and a vector product, which Gibbs felt was “superior to those of Hamilton.”[55]

Although Gibbs admitted that “I am not …conscious that Grassmann’s writings exerted any particular influence on my [Vector Analysis]” Gibbs decided to add a little introductory paragraph to his pamphlet in 1888 where he specifically mentioned Clifford and Grassmann, even though Clifford seemed to have had minor influence and Grassmann not at all.[56] Gibbs wrote his friend that he was “glad enough in the introductory paragraph to shelter myself behind one or two distinguished names in making changes of notation which I felt would be distasteful to quaternionists… [even though] it all follows with the inexorable logic of algebra from the problem which I had set myself long before my acquaintance with Grassmann.”[57]

Gibbs had vector algebra right there, with some big names to back him up, and he printed it but he delayed and delayed publishing it, “due to difficulty in making up my mind in respect to details of notation.”[58]

Meanwhile, in England, a former telegraph operator came to the same idea, which brings us, finally, to.

Part 5: Oliver Heaviside (1873-87)

Oliver Heaviside’s life was transformed by reading Maxwell’s 1873 book. Years later he recalled, “When I saw on the table in the library the work that had just been published, I browsed through it and I was astonished! …I saw that it was great, greater and greatest, with prodigious possibility in its power.” However, Heaviside had only part of a high school education (he had to leave school at 16) and he recalled that he only “learned only school algebra and trigonometry, which I had largely forgotten.” [59]

The following year, Heaviside quit his job working on telegraphs and moved back in with his parents where he focused on his theoretical and mathematical research. There he published paper after paper on the mathematical theories of telegraphs derived from his blossoming understanding of Maxwell’s equations. Supposedly he worked mostly in the middle of the night with all the doors and windows closed and the fires burning “hotter than hell.”[60]

Starting in 1882, at the request of the editor, Heaviside began to publish in a weekly trade journal called The Electrician. In November of 1882, Heaviside published his first paper on the mathematics of the magnetic field from a current carrying wire using vectors where he used Maxwell’s term of “curl” in isolation without using the vector result of a quaternion.[61]

By December 2, 1882, Heaviside decided that although quaternions are, “a very remarkable system of mathematics” which he said were superficially attractive as, “one equation takes the place of three.” Heaviside decided that, “the operations met with are much more difficult than the corresponding ones in the ordinary system, so that the saving of labour is, in a great measure, imaginary.” Heaviside therefore suggested that they should, “make a compromise; look behind the easily managed but complex scalar equations, and see the single vector one behind them, expressing the real thing.”[62]

By April of the following year, 1883, Heaviside seemed to have independently conceived of the idea of divergence as Clifford did 5 years before writing, “[as] the expression… with the negative sign prefixed Maxwell called the ‘convergence’… we may as well use the term ‘divergence’ for the same quantity with the + sign prefixed”[63] (Sorry that I attributed this nomenclature to Heaviside in my Hamilton video, I didn’t realize that Clifford made that name first).

Anyway, in June 1885, Heaviside decided that having most of his equations written in cartesian form and only using the shorthand of curl and divergence where that occurred was not sufficient. As he put it, “owing to the extraordinary complexity of the investigation when written out in Cartesian form (which I began doing, but gave up aghast), some abbreviated method of expression becomes desirable.”[64]

Then, like Gibbs before him, Heaviside described two types of vector multiplication: a “scalar product” and “vector product.”[65] However, unlike Gibbs who used the modern notations of a dot for the scalar multiplication and a “x” for the vector multiplication, Heaviside used no term to be the scalar result and a “V” to represent the vector result. A big problem with this method is the confusion with what \nabla N represents. If “N” is a scalar “p” than, like with modern vector algebra, \nabla p is the gradient of p which is a vector. On the other hand, if “N” is a vector “A” than \nabla A is the divergence of A which results in a scalar.

It was for this reason that the following year, that Heaviside realized that his notation was confusing and promoted the idea of using a different font, the “Clarendons” font that looks bold to represent vectors.[66]  But even with this change, it was still confusing.

Anyway, years later, Heaviside said that Tait contacted him after this paper and he “appeased Tait considerably (during a little correspondence we had) by disclaiming any idea of discovering a new system.” Heaviside’s words apparently mollified the invested Tait, especially as, as Heaviside wryly recalled, “I was too small to be seen, at first.”[67] Heaviside wasn’t kidding about being “small at first,” basically no one was particularly interested in what he had to say. In 1887, Heaviside was told that the new owner of the magazine The Electrician wanted him to discontinue his work on Maxwell’s equations altogether. Heaviside sadly noted that the owner told him that, “he had made particular enquiries amongst students who would be likely to read my papers to find if anyone did so, he had been unable to discover a single one.”[68]

Heaviside must have been desperate, but just then an event happened which converted almost everyone into a Maxwellian. That event? A little discovery of an invisible wave by a young German scientist named Heinrich Hertz.

Part 6: Hertz changes the game (1887-1890)

That same year that Heaviside was told that no one was interested in his mathematics on Maxwell’s equations, 1887 a 30 year old German scientist named Heinrich Hertz published a paper titled “On Very Rapid Electric Oscillations.” This is significant because this was the paper where Hertz demonstrated that he could create an invisible electromagnetic wave, that we now call radio wave, that moves at the speed of light from vibrating electronics. Hertz wasn’t motivated by wireless telegraphy, instead, according to Hertz, “the object of these experiments was to test the fundamental hypotheses of the Faraday-Maxwell theory.”[69] In other words, Hertz’s motivation was to experimentally demonstrate as Faraday and Maxwell had postulated that light itself was a vibrating electromagnetic wave.

Suddenly, there was incredible interest in Maxwell’s equations, but, as most people didn’t know quaternions, Maxwell’s “elementary” book just frightened them away. Heaviside wasted no time with the renewed interest and by February, 1888, published the first of many papers “on electromagnetic waves” for the prestigious Philosophical Magazine, where he mentioned his previous papers from a few years earlier in The Electrician.[70] Soon several important people began to read and reference Heaviside. For example, in March, 1889, the Maxwellian Oliver Lodge gushed in a footnote: “of one whose name is not yet on everybody’s lips, but whose profound researches into electro-magnetic waves have penetrated further than anybody yet understands into the depths of the subject, and whose papers have very likely contributed largely to the theoretical inspiration of Hertz – I mean that powerful mathematical physicist, Mr. Oliver Heaviside.”[71]

Now it turns out that Lodge was incorrect, in that Hertz had not read Heaviside before he published his experimental paper on radio waves. However, Hertz did read Heaviside’s 1888 theoretical paper after his published his experiment and was intrigued by it. Soon, Hertz and Heaviside started a correspondence, where they bonded over the belief that Maxwell shouldn’t have focused so much on electric and magnetic potentials, and instead believed, as Heaviside dramatically put it in January of 1889, “best to murder the whole lot.”[72] In March of 1890, Hertz published his own paper “On the Fundamental Equations of Electro-Magnetics” where he admitted, “Mr. Oliver Heaviside has been working in the same direction [as me] ever since 1885,” which he learned from “the Philosophical Magazine for February 1888.”[73] (Note that some people think that Hertz was involved with the development of vector algebra due to this comment, but Hertz was actually referring to their shared dislike of potentials, and this paper does not include vectors in any form, although later he did attempt to add a bit in later papers due to Heaviside’s influence).

Meanwhile, Gibbs, who still hadn’t published his pamphlet on vector analysis, had been inspired by Hertz’s discovery of 1887 to distribute his pamphlet around. It is unclear to me if Tait was directly sent a pamphlet, or if he got it second hand, but either way, he didn’t like it. Which leads us to.

Part 7: The War of the Vectors (1890-1894)

Possibly inspired by Hertz’s results, Peter Tait began to work on writing a third edition “much enlarged” of his “Elementary Treatise of Quaternions.” However, his frustration with the slow acceptance of quaternions was starting to grind him down. Gibbs’ work, in particular, infuriated Tait. Tait had known Williard Gibbs for many years through his work with Maxwell on thermodynamics and knew as well as anyone that a man as smart as Gibbs could handle the mathematics of quaternions.

In frustration, Tait snapped and added in the introduction that “it is disappointing to find how little progress has recently been made with the development of Quaternions.” And that, “Even Prof Willard Gibbs must be ranked as one of the retarders of quaternion progress, in virtue of his pamphlet on ‘Vector Analysis,’ a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann.”[74] Note that people born intersex are not monsters and anyone being cruel about human beings who are born different in the comments will be banned. Sorry, but this is my channel and there is enough intolerance in the world, and I cannot have my channel add to it.

Anyway, that line, especially with the vulgar and mean-spirited word, (which Tait probably meant as a joke) grabbed the attention of many people including Gibbs who responded in Nature magazine on April 2, 1891. In this retort, Gibbs noted that if the issue was his notation, that the objection should not be that he was “monstrous” but instead more that “its dress as uncouth.” Gibbs then tried to explain that although, “the quaternion affords a convenient notation for rotations… [that using] \alpha \bullet \beta and \alpha \times \beta for what is expressed in quaternions by -S \alpha \beta and V \alpha \beta , and in a like manner \nabla \bullet \alpha and \nabla \times \alpha for -S\nabla \alpha and V\nabla \alpha in quaternions… have some substantial advantages over the quaternionic in point of convenience.” [75] [By the way, how prescient was Gibbs? That is exactly, I mean EXACTLY what most physicists believe today.]

Then in June, Oliver Heaviside added fuel to the fire in an article titled “on the Forces, Stresses and Fluxes of Energy.” In this paper, Heaviside stood up and added that “I rejected the quaternionic basis of vector-analysis” and that “Hamilton’s system” was “metaphysics” (although Heaviside still rejected Gibb’s notation, despite the fact that Gibbs showed him his obviously superior notation three years earlier in 1888.) In this paper, Heaviside was still moderately polite about quaternions writing, “the modifications I made…[are] simply the elements of Quaternions without the quaternions, with the notation simplified to the uttermost, and with the very inconvenient minus sign before scalar products done away with.”[76]

However, by November, Heaviside was invited back to The Electrician where he done with niceties and declared, “I came later to see that, so far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude”[77]

Soon, everyone was printing nasty things about each other and no one seemed to realize that not only were both vector algebra and quaternions both useful, but that they complimented each other. Vector algebra was derived from quaternions and then, in return, vector algebra is a nearly vital first step towards understanding quaternions (if your name is not Hamilton, Tait, Maxwell, Gibbs, Heaviside or a scant handful of others.) Not to mention that Maxwell’s equations are easier to learn and use with vector algebra and honestly, we need all the help we can get.

Then, on January 1, 1894, the scientific community was horrified to learn that Heinrich Hertz, who was only 36 years old, had died of some mysterious ailment. In June in England, Oliver Lodge put together a talk on the life and work of his newly found friend,[78] a talk where he changed Hertz’s receiver so that it could be demonstrated for a crowd which initiated the wireless revolution,  which of course increased interest in Maxwell’s equations. In Germany it seemed as if the whole scientific community went into mourning. After that, many scientists (especially in Germany) turned to Oliver Heaviside as their representative of Maxwell’s equations, after all the great Hertz had recommended him.

Part 8: Tait loses the war (1894-1901)

That same year that Hertz passed, a German scientist named August Föppl finished his influential book on electricity where he wrote that, “The works of [Heaviside] influenced my presentation more than those of any other physicist with the obvious exception of Maxwell himself. I consider Heaviside to be the most eminent successor to Maxwell in regards to theoretical developments, just as it was Hertz – who alas was so quickly snatched from us – who was Maxwell’s most eminent successor in regard to experimental developments.”[79] Föppl also acknowledged that an alternative motivation for promoting Heaviside and his vector analysis is because it resembled the mathematics of the German mathematician Grassmann and “The country which produced a Grassmann should no longer stand behind the country of Hamilton with the introduction of this important improvement in the mathematical means of theoretical physics.”[80]

Meanwhile, in England, Heaviside published a book titled  “Electromagnetic Theory” which, along with Föppl’s book, converted most people to Heaviside’s view of Maxwell’s equations and his view of vectors. Soon, Grassmann’s influence was exaggerated and quaternions were dismissed as evil or at least distasteful.

You can see Heaviside’s influence from this book in many of the names and conventions used in modern Maxwell’s equations. For example, Heaviside created the name “permittivity” in 1887, and restated it in his 1893 book and it is still in use today.[81] Heaviside also promoted the use of the term “voltage” which was used previously but rarely to replace the common term of “pressure.” Amusingly, in 1893 he wrote how irritated he was by “the utterly vicious misuse of pressure to indicate EMF or voltage, by men who are old enough to know better and do.” [82]

As a reviewer of Heaviside’s 1893 book named George Minchin noted:  “As everyone knows, Mr. Heaviside has advocated many changes in scientific nomenclature, and has already succeeded in getting some of them adopted.” Also, it was in the same review of Heaviside’s book when Minchin noted how difficult it would be to hand write the “Clarendon” type, and that Heaviside’s suggestion of a suffix would sometimes result in a double suffix and suggested “Perhaps a horizontal bar over the letters is the best, though this is not good.”[83]  Quickly, Minchin’s tentative suggestion of a bar was replaced with an arrow, which it has remained ever since.

After Föppl’s and Heaviside’s books, Heaviside was at the height of his fame and influence, but he instead became fascinated with geophysics and meteorology and didn’t do much to influence the development of Maxwell’s laws or vector analysis after 1900.[84] Even after Heaviside’s contributions were largely forgotten for a time, his insistence on the “evil” of quaternions remained.

Meanwhile, Gibbs had moved back to his research on statistical mechanics and also did not do much more on Maxwell’s equations or vector algebra. This was particularly relevant in early 1901 when Gibbs was approached by the administrators of his university to turn his vector pamphlet into a book in connection with Yale’s bicentennial. Gibbs was too busy with his statistical mechanics but was delighted when a former student named Edwin Wilson agreed to take a stab at it.[85]

In the introduction, Wilson said that although the “greater part of the material used in the following pages has been taken from the course of lectures on Vector Analysis …by Professor Gibbs,” he also used Oliver Heaviside’s 1893 book and August Föppl’s 1894 book and that his “previous study of Quaternions has also been of great assistance.”[86]

You can see how popular Heaviside was at the time as Wilson justified the need for his book by quoting Heaviside’s 1893 book where Heaviside said that the work he had written would “serve as a stopgap till regular vectorial treatises come to be written suitable for physicists.”[87]

Wilson then made several minor changes that were to have lasting impact on modern mathematics due their deeply thought clarity of instruction and learning. First, Wilson repeated Heaviside’s idea of a different font, the “Clarendon type,” for vectors so that “it [is] possible to pass from directed quantities to their scalar magnitude by a mere change in the appearance of a letter without any confusing change in the letter itself.” (like “v [for] the velocity of a moving mass, [and] v for the magnitude of that velocity.”[88]) In this way Wilson realized that even when the multiplication displays whether a variable was a vector or a scalar it was still a significant help to have a quick way to tell the nature of a variable by its look (either bold in print or with an arrow if on paper).

Second, Wilson moved up the dot in the scalar product so that it wouldn’t be confused with a period, and noted that as you read the scalar multiplication as A dot B, scalar multiplication “may often be called the dot product instead of the direct product.”[89]

Third, and you probably guessed this one, Wilson also noted that the vector multiplication is read “A cross B. For this reason, it is often called the cross product.”[90]

Finally, Wilson was the one who came up with the name del for Hamilton’s triangle operator. His logic was that “This symbolic operator \nabla was introduced by Sir William Rowan Hamilton and is now in universal employment. There seems, however, to be no universally recognized name for it” Adding in a footnote that some use Nabla, some use Alted, and Föppl called it “die Operation \nabla ” causing Wilson to note dryly “How this is to be read is not divulged.”

However, Wilson felt strongly that “for lecturing and purposes of instruction something is required – something too that does not confuse the speaker or hearer even when often repeated.” It was for that reason that Wilson decided that “the monosyllable del is so short and easy to pronounce that even in complicated formulae in which  occurs a number of times no inconvenience to the speaker or hearer arises from the repetition.”[91] And that, ladies and gentlemen, is why I call it del.

One more comment about the del (or Nabla or alted) operator. There are a million debates about Hamilton vs. Grassmann vs. Clifford and the history of vector algebra, but no one can debate that Hamilton was the one who created the del function. This function can be used to represent the Laplacian, the gradient, the divergence and the curl. This is the function that attracted Tait, and it was Tait’s work on it that appealed to Maxwell which is why both Heaviside and Gibbs created vector algebra. Could you make vector algebra from Grassmann? Of course. Would we if we didn’t have the del operator? Doesn’t seem likely to me, and, also, it wasn’t what happened.

Meanwhile, just before Wilson published the first book on Vector Analysis, on July 4, 1901, Tait passed away at the age of 70. William Thomson, who was then known as Lord Kelvin wrote his obituary and ended it with a personal note that Tait’s “death is a loss to me which cannot, as long as I live, be replaced.”[92] Without Tait’s push, the quaternions were eventually abandoned and only semi-recently have been revived, just as Gibbs foretold, mostly as a method for computing rotations.

Now you might notice that I didn’t mention how Heaviside combined or formulated 4 modern Maxwell’s equations. That is because he didn’t. Heaviside, like Maxwell before him, mentioned the 4 equations that we now consider “Maxwell’s equations” in his papers between 1884 and 1885, but didn’t cause anyone to consider Maxwell to have only 4 main equations. Heck, in his influential 1893 book, Heaviside was considered to have 8 equations, including terms for a magnetic density and magnetic current even though they were, according to Heaviside himself “fictitious”[93]

So, if Maxwell didn’t cause people to think he had 4 equations, and Heaviside didn’t either, who did? The first person I can find who emphasized the 4 equations of Maxwell’s was not Maxwell or Heaviside or even Gibbs or Hertz but actually Hendrik Lorentz in 1895.[94]

In fact, I contend that it was only due to the popularity of relativity that was derived from Lorentz’s ideas that people started to think of Maxwell having 4 equations.[95]  My theory of how we got to the 4 modern Maxwell’s equations and why so many falsely think it was Heaviside is next time on the evolution of wireless.

[1] Tait is shy about the cruel nickname, calling it “a not very complimentary nickname” Tait “James Clerk Maxwell” (1880) Proceedings of the Royal Society of Edinburgh vol. 10  p. 332 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/08663EBC2051BB8977BABC1A6442F138/S0370164600052299a.pdf/james_clerkmaxwell.pdf but it is spelled out in https://www.google.com/books/edition/The_Christian_miscellany_and_family_visi/1TMEAAAAQAAJ?hl=en&gbpv=0

[2] Tait “James Clerk Maxwell” (1880) Proceedings of the Royal Society of Edinburgh vol. 10  p. 332 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/08663EBC2051BB8977BABC1A6442F138/S0370164600052299a.pdf/james_clerkmaxwell.pdf

[3] Cargill Knott Life and Scientific Work of Peter Guthrie Tait vol. 1 (1911) p. 7 https://www.google.com/books/edition/Life_and_Scientific_Work_of_Peter_Guthri/P-I9AAAAMAAJ?hl=en&gbpv=0

[4] Lewis Campbell The Life of James Clerk Maxwell (1882) p. 106 https://www.google.com/books/edition/The_Life_of_James_Clerk_Maxwell/lJfvAAAAMAAJ?hl=en&gbpv=0

[5] Tait’s notebook quoted in Cargill Knott Life and Scientific Work of Peter Guthrie Tait vol. 1 (1911) p. 8 https://www.google.com/books/edition/Life_and_Scientific_Work_of_Peter_Guthri/P-I9AAAAMAAJ?hl=en&gbpv=0

[6] Maxwell to Campbell (Nov 9, 1851) The Scientific Letters and Papers of James Clerk Maxwell vol. 1 p. 208 https://www.google.com/books/edition/The_Scientific_Letters_and_Papers_of_Jam/zfM8AAAAIAAJ?hl=en&gbpv=0

[7] Maxwell to Thomson (Feb 20, 1854) The Scientific Letters and Papers of James Clerk Maxwell vol. 1 p. 237 https://www.google.com/books/edition/The_Scientific_Letters_and_Papers_of_Jam/zfM8AAAAIAAJ?hl=en&gbpv=0

[8] Maxwell to Thomson (Feb 20, 1854) 1901) Silvanus Thompson The Life of William Thomson, Baron Kelvin vol. 1 (1910) p. 304-5 https://www.google.com/books/edition/The_Life_of_William_Thomson_Baron_Kelvin/vfLPAAAAMAAJ?hl=en&gbpv=

[9] Maxwell, A Treatise on Electricity and Magnetism Vol 1, (1873) p. x https://www.google.com/books/edition/A_Treatise_on_Electricity_and_Magnetism/eSNZAAAAcAAJ?hl=en&gbpv=0

[10] Maxwell, “On Faraday’s Lines of Force” (1856) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 162 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0

[11] Maxwell, James Clerk “On Faraday’s Lines of Force” (read Dec 1855, Feb 1856) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 180-1 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0

[12] Maxwell, James Clerk “On Faraday’s Lines of Force” (read Dec 1855, Feb 1856) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 164 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0

[13] Maxwell, James Clerk “On Faraday’s Lines of Force” (read Dec 1855, Feb 1856) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 172 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0

[14] Hamilton “On Quaternions (cont.)” The London and Edinburgh Phil Magazine vol 31 (1847) p. 290-1

[15] Cargill Knott Life and Scientific Work of Peter Guthrie Tait vol. 1 (1911) p. 12 https://www.google.com/books/edition/Life_and_Scientific_Work_of_Peter_Guthri/P-I9AAAAMAAJ?hl=en&gbpv=0

[16] Andrews to Hamilton (Aug 11, 1858) found in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait vol. 1 (1911) p. 119 https://www.google.com/books/edition/Life_and_Scientific_Work_of_Peter_Guthri/P-I9AAAAMAAJ?hl=en&gbpv=0

[17] Hamilton “On Quaternions” The London and Edinburgh Phil Magazine vol 29 (1846) p. 26 https://www.google.com/books/edition/The_London_Edinburgh_and_Dublin_Philosop/W-K3nzGdhhwC?hl=en&gbpv=0

[18] Hamilton “On Quaternions (cont.)” The London and Edinburgh Phil Magazine vol 31 (1847) p. 290-1

[19] (Note that Hamilton varied the orientation of his triangle in different papers). Hamilton Lectures on Quaternions (1853) p. 611  https://www.google.com/books/edition/Lectures_on_Quaternions/PJIKAAAAYAAJ

[20] (Note that Hamilton varied the orientation of his triangle in different papers). Hamilton Lectures on Quaternions (1853) p. 611  https://www.google.com/books/edition/Lectures_on_Quaternions/PJIKAAAAYAAJ

[21] Tait to Hamilton (Aug 19, 1858), found in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait vol. 1 (1911) p. 119 https://www.google.com/books/edition/Life_and_Scientific_Work_of_Peter_Guthri/P-I9AAAAMAAJ?hl=en&gbpv=0

[22] Tait to Hamilton (Aug 19, 1858), found in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait vol. 1 (1911) p. 120 https://www.google.com/books/edition/Life_and_Scientific_Work_of_Peter_Guthri/P-I9AAAAMAAJ?hl=en&gbpv=0

[23] Hamilton to Tait (April 12, 1859) found in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait vol. 1 (1911) p. 134 https://www.google.com/books/edition/Life_and_Scientific_Work_of_Peter_Guthri/P-I9AAAAMAAJ?hl=en&gbpv=0

[24] Hamilton to Tait (May 27, 1861) Graves The Life of Sir William Rowan Hamilton vol. 3 (1889) p. 133 https://archive.org/details/lifesirwilliamr01gravgoog/page/132/mode/2up

[25] According to a contemporary Maxwell was “one of the remarkable men known to the scientific world” but “the power of oral exposition… [and] it was the deficiency of this power in Professor Maxwell principally that made the curators prefer Mr Tait” in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait (2015) p. 16 Life and Scientific Work of Peter Guthrie Tait – Google Books

[26] Thomson to “James” (Feb 14, 1860) Silvanus Thompson The Life of William Thomson, Baron Kelvin vol. 1 (1910) p. 408 https://www.google.com/books/edition/The_Life_of_William_Thomson_Baron_Kelvin/vfLPAAAAMAAJ?hl=en&gbpv=0

[27] Thomson to David King (Jan 8, 1862) Silvanus Thompson The Life of William Thomson, Baron Kelvin vol. 1 (1910) p. 422 https://www.google.com/books/edition/The_Life_of_William_Thomson_Baron_Kelvin/vfLPAAAAMAAJ?hl=en&gbpv=0

[28] Maxwell to Tait (March 7, 1865) The Scientific Letters and Papers of James Clerk Maxwell (1990) p. 214

[29] Tait, Peter An Elementary Treatise on Quaternions (1867)p. v-vi https://www.google.com/books/edition/An_Elementary_Treatise_on_Quaternions/v6ANAAAAQAAJ?hl=en&gbpv=1&kptab=overview

[30] Thomson to Prof Chrystal (July 13, 1901) Silvanus Thompson The Life of William Thomson, Baron Kelvin vol. 1 (1910) p. 452 https://www.google.com/books/edition/The_Life_of_William_Thomson_Baron_Kelvin/vfLPAAAAMAAJ?hl=en&gbpv=0

[31] Thomson “Memorial: Peter Tait” (1901) Silvanus Thompson The Life of William Thomson, Baron Kelvin vol. 1 (1910) p. 479 https://www.google.com/books/edition/The_Life_of_William_Thomson_Baron_Kelvin/vfLPAAAAMAAJ?hl=en&gbpv=0

[32] Maxwell to Tait (Dec 11, 1867) The Scientific Letters and Papers of James Clerk Maxwell (1990) p. 330 https://archive.org/details/scientificletter0000maxw/page/330/mode/2up

[33] Tait, Peter An Elementary Treatise on Quaternions (1867)p. 267 An elementary treatise on quaternions – Google Books

[34] Tait, Peter An Elementary Treatise on Quaternions (1867)p. vi An elementary treatise on quaternions – Google Books

[35] Maxwell to Campbell (Jan 31, 1858) found in Campbell, Lewis and Garnett, William The Life of James Clerk Maxwell (1882) p. 301 https://www.google.com/books/edition/The_Life_of_James_Clerk_Maxwell/a_xAAQAAIAAJ?hl=en&gbpv=0

[36] Campbell, Lewis and Garnett, William The Life of James Clerk Maxwell (1882) p. 276 https://www.google.com/books/edition/The_Life_of_James_Clerk_Maxwell/a_xAAQAAIAAJ?hl=en&gbpv=0

[37] Maxwell “On Physical Lines of Force: Part III” (read Jan, Feb, 1862) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) Gauss’s Law (equation 115 p. Ampere’s Law with Maxwell’s addition (equation 112, p. 496)  Gauss’s Law for Magnets (equation 56  p. 476) Faraday’s law (equation 54 p. 475) https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0

[38] Weber and Kohlrausch “Reduction of the Measures of Intensity of Currents to Mechanical Units” found in the Appendix of Auguste de la Rive “A Treatise on Electricity” (1858) p. 747 https://www.google.com/books/edition/A_Treatise_on_Electricity/yo9PAAAAYAAJ?

[39] Maxwell “On Physical Lines of Force: Part III” (read Jan, Feb, 1862) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 500 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0

[40] Maxwell “A Dynamical Theory of the Electromagnetic Field” (read Oct, Dec 1864) The Scientific Papers of James Clerk Maxwell vol. 1 (1890) p. 535-6 https://www.google.com/books/edition/The_Scientific_Papers_of_James_Clerk_Max/edlCAQAAMAAJ?hl=en&gbpv=0

[41] Maxwell to Charles Tayler (Feb 2, 1866) The Scientific Letters and Papers of James Clerk Maxwell vol. 2 (1990) p. 249 https://archive.org/details/scientificletter0000maxw/page/248/mode/2up

[42] Tait to Maxwell (Nov 27, 1867) which is in a footnote to the letter from Maxwell to Tait (Dec 4, 1867) found in The Scientific Letters and Papers of James Clerk Maxwell vol. 2 (1990) p. 323 https://archive.org/details/scientificletter0000maxw/page/n369/mode/2up

[43] Maxwell to Tait (Nov 7, 1870) found in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait (2015) p. 143-4 Life and scientific work of Peter Guthrie Tait

[44] Maxwell to Tait (Nov 7, 1870) found in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait (2015) p. 143-4 Life and scientific work of Peter Guthrie Tait

[45] Maxwell to Lewis Cambell (Oct 19, 1872) found in Cambell, Lewis and Garnett, William The Life of James Clerk Maxwell (1882) p. 383-4 https://www.google.com/books/edition/The_Life_of_James_Clerk_Maxwell/a_xAAQAAIAAJ?hl=en&gbpv=0

[46] Maxwelll to Tait (Jan 23, 1871) found in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait (2015) p. 145 Life and scientific work of Peter Guthrie Tait

[47] Maxwell, James Clerk A Treatise on Electricity and Magnetism: vol. 1 (1873) p. 28-9 https://www.google.com/books/edition/A_Treatise_on_Electricity_and_Magnetism/gokfAQAAMAAJ?hl=en&gbpv=1&dq=James+Clerk+Maxwell+A+Treatise+on+Electricity+and+Magnetism:+vol.+1&printsec=frontcover

[48] Lewis Campbell The Life of James Clerk Maxwell (1884) p. 319 The Life of James Clerk Maxwell – Google Books

[49] Colin Mackensie quoted in Lewis Campbell The Life of James Clerk Maxwell (1884) p. 322 The Life of James Clerk Maxwell – Google Books

[50] Tait “Clerk Maxwell’s Scientific Work” (Feb 5, 1880) Nature Magazine vol. 21 p. 321 https://www.google.com/books/edition/Nature/eSQ6AQAAMAAJ?hl=en&gbpv=0

[51] Gibbs to Victor Schlegel (1888) found in Crowe A History of Vector Analysis (1994) p. 152 https://www.google.com/books/edition/A_History_of_Vector_Analysis/iVFAVqA91h4C?hl=en&gbpv=0

[52] Gibbs, Josiah Elements of Vector Analysis (1884) p. 5  https://www.google.com/books/edition/Elements_of_Vector_Analysis/VurzAAAAMAAJ?hl=en&gbpv=1

[53] William Clifford, Elements of Dynamic: Part I (1878) p. 209 https://www.google.com/books/edition/Elements_of_Dynamic/sbQEAAAAYAAJ?hl=en&gbpv=0

[54] Gibbs, Josiah Elements of Vector Analysis (1884) p. 16-7 https://www.google.com/books/edition/Elements_of_Vector_Analysis/VurzAAAAMAAJ?hl=en&gbpv=1

[55] Gibbs to Victor Schlegel (1888) found in Crowe A History of Vector Analysis (1994) p. 153 https://www.google.com/books/edition/A_History_of_Vector_Analysis/iVFAVqA91h4C?hl=en&gbpv=0

[56] Gibbs, Josiah Elements of Vector Analysis (1884) p. 1 https://www.google.com/books/edition/Elements_of_Vector_Analysis/VurzAAAAMAAJ?hl=en&gbpv=1

[57] Gibbs to Victor Schlegel (1888) found in Crowe A History of Vector Analysis (1994) p. 153 https://www.google.com/books/edition/A_History_of_Vector_Analysis/iVFAVqA91h4C?hl=en&gbpv=0

[58] Gibbs to Victor Schlegel (1888) found in Crowe A History of Vector Analysis (1994) p. 153 https://www.google.com/books/edition/A_History_of_Vector_Analysis/iVFAVqA91h4C?hl=en&gbpv=0

[59] Heaviside (Feb 24, 1918) quoted in Paul J. Nahin Oliver Heaviside: The Life, Work and Times (2002) p. 24-5 https://www.google.com/books/edition/Oliver_Heaviside/e9wEntQmA0IC?hl=en&gbpv=0

[60] Quoted in Vladimir Naumovich Zharkov Interior Structure of the Earth (1986) p. 43 Interior structure of the earth and planets – Google Books

[61] Heaviside, Oliver “The Relations between Magnetic Force and Electric Current” (Nov 18, 1882) The Electricians vol. 10 p. 6-7 https://www.google.com/books/edition/_/qmlNAAAAYAAJ?gbpv=0

[62] Heaviside, Oliver “Connected General Theorems in Electricity and Magnetism” (Dec 2, 1882) The Electrician vol. 10 p. 55 https://www.google.com/books/edition/_/qmlNAAAAYAAJ?gbpv=0

[63] Heaviside, Oliver “Some Electrostatic & Magnetic Relations” (April 14, 1883) The Electrician vol. 10 p. 510 https://www.google.com/books/edition/The_Electrician/uRhbAAAAYAAJ?hl=en&gbpv=1&bsq=Heaviside%20

[64] Heaviside “On the Electromagnetic Wave Surface” (June 1885) The London, Edinburgh and Dublin Phil Magazine 5th Series p. 399 https://www.google.com/books/edition/The_London_Edinburgh_and_Dublin_Philosop/QXvkAAAAMAAJ?hl=en&gbpv=0

[65] Heaviside “On the Electromagnetic Wave Surface” (June 1885) The London, Edinburgh and Dublin Phil Magazine 5th Series p. 401 https://www.google.com/books/edition/The_London_Edinburgh_and_Dublin_Philosop/QXvkAAAAMAAJ?hl=en&gbpv=0

[66] Heaviside Electromagnetic Theory (1893) p. 141 Electromagnetic Theory – Google Books

[67] Heaviside, “Reviews: Vector Analysis” (March 21, 1902) The Electrician vol. 48 p. 862 https://www.google.com/books/edition/The_Electrician/wpc9AQAAMAAJ?hl=en&gbpv=0

[68] E. T. Whittaker, “Oliver Heaviside” in Calcutta Mathematical Society Bulletin vol. 20 (1928) p. 209 Commeration Volume on the occasion of the Twentieth Anniversary of the foundation of the Calcutta Mathematical Society in 1928: Bulletin, Vol-20 : Free Download, Borrow, and Streaming : Internet Archive [Note that the author did not read Maxwell’s 1861-2 papers or his 1864 papers and therefore mistakenly thought that Heaviside created Maxwell’s 4 modern equations]

[69] Hertz “Introduction” Electric Waves (1900) p. 20 Electric Waves – Google Books

[70] Heaviside “On Electromagnetic Waves” The London, Edinburgh and Dublin Philosophical Magazine (1888) p. 130 https://www.google.com/books/edition/_/kRk0nZ_w6oUC?hl=en&gbpv=0

[71] Oliver Lodge “The Discharge of a Leyden Jar” Nature (March 14, 1889) p. 473 Nature

[72] Heaviside, “The General Solution of Maxwell’s Electromagnetic Equations” Heaviside Electromagnetic Waves (1889) p. 45 https://www.google.com/books/edition/Electromagnetic_Waves/4pM3AAAAMAAJ?hl=en&gbpv=0

[73] Heaviside “On the Fundamental Equations of Electro-Magnetics” (March 19, 1890) translated and found in Hertz Electric Waves (1962) p. 196 https://www.google.com/books/edition/Electric_Waves/8GkOAAAAIAAJ?hl=en&gbpv=0

[74] Tait An Elementary Treatise on Quaternions (1890) p. vi https://www.google.com/books/edition/An_Elementary_Treatise_on_Quaternions/CGZLAAAAMAAJ?hl=en&gbpv=0

[75] Gibbs “On the Role of Quaternions in the Algebra of Vectors” (April 2, 1891) p. 512 https://www.google.com/books/edition/_/vz_V5qQXAGoC?gbpv=0

[76] see the footnote Heaviside, Oliver “On the Forces, Stresses and Fluxes of Energy” (June, 1891) Phil Transactions of the Royal Society of London vol. 183, (Dec 1892) p. 428 XI. On the forces, stresses, and fluxes of energy in the electromagnetic field (royalsocietypublishing.org)

[77] Heaviside “Electromagnetic Theory – XIX” (Nov 13, 13, 1891 The Electrician vol. 28 p. 27 https://www.google.com/books/edition/The_Electrician/85cvAAAAYAAJ?hl=en&gbpv=0

[79] Föppl Einführung in die maxwell’sche Theorie der Elektricität (1894) p. vii https://www.google.com/books/edition/Einführung_in_die_maxwell_sche_Theorie/93dKAAAAMAAJ?hl=en&gbpv=0 quoted and translated by Michael Crowe, History of Vector Analysis (1994) p. 176 https://www.google.com/books/edition/A_History_of_Vector_Analysis/iVFAVqA91h4C?hl=en&gbpv=0

[80] Föppl Einführung in die maxwell’sche Theorie der Elektricität (1894) p. vii https://www.google.com/books/edition/Einführung_in_die_maxwell_sche_Theorie/93dKAAAAMAAJ?hl=en&gbpv=0 quoted and translated by Michael Crowe, History of Vector Analysis (1994) p. 176 https://www.google.com/books/edition/A_History_of_Vector_Analysis/iVFAVqA91h4C?hl=en&gbpv=0

[81] Heaviside “Electromagnetic Induction: Nomenclature Scheme” (June 17, 1887) The Electrician vol. 19 p. 124  The Electrical Journal – Google Books

[82] Heaviside Electromagnetic Theory (1893) p. 109 Electromagnetic Theory – Google Books

[83] G. M. Minchin “Notices respecting New Books” Philosophical Magazine  vol. 38 (1894) p. 148 Philosophical Magazine – Google Books

[84] According to  E. T. Whittaker, “Oliver Heaviside” in Calcutta Mathematical Society Bulletin vol. 20 (1928) p. 217-8 Commeration Volume on the occasion of the Twentieth Anniversary of the foundation of the Calcutta Mathematical Society in 1928: Bulletin, Vol-20 : Free Download, Borrow, and Streaming : Internet Archive

[85] Edwin Wilson Vector Analysis (1901), p. ix https://archive.org/details/117714283/page/n17/mode/2up

[86] Edwin Wilson Vector Analysis (1901), p. vi https://archive.org/details/117714283/page/n17/mode/2up

[87] Heaviside quoted by Edwin Wilson Vector Analysis (1901) p. xii https://archive.org/details/117714283/page/n21/mode/2up

[88] Edwin Wilson Vector Analysis (1901) p. 4 https://archive.org/details/117714283/page/4/mode/2up

[89] Wilson, Edwin Vector Analysis (1901), p. 55 https://archive.org/details/117714283/page/54/mode/2up

[90] Wilson, Edwin Vector Analysis (1901), p. 61 https://archive.org/details/117714283/page/60/mode/2up

[91] Wilson, Edwin Vector Analysis (1901), p. 138 https://archive.org/details/117714283/page/138/mode/2up?q=del+

[92] Thomson “Memorial: Peter Tait” (1901) Silvanus Thompson The Life of William Thomson, Baron Kelvin vol. 1 (1910) p. 479 https://www.google.com/books/edition/The_Life_of_William_Thomson_Baron_Kelvin/vfLPAAAAMAAJ?hl=en&gbpv=0

[93] Heaviside Electromagnetic Theory (1893) p. 50 Electromagnetic Theory – Google Books

[94] Lorentz “”ELECTRISCHEN UND OPTISCHEN ERSCHEINUNGEN IN BEWEGTEN KÖRPERN” (1895) p. 16-17 https://www.google.com/books/edition/Versuch_einer_Theorie_der_electrischen_u/eZHOAAAAMAAJ?hl=en&gbpv=1&dq=Lorentz+%22ELECTRISCHEN+UND+OPTISCHEN+ERSCHEINUNGEN+IN+BEWEGTEN+K%C3%96RPERN%22&printsec=frontcover

[95] Tolman, The Theory of Relativity of Motion (1917) p. 16 https://www.google.com/books/edition/The_Theory_of_the_Relativity_of_Motion/W6EIAAAAIAAJ?hl=en&gbpv=1