Quaternions Are Amazing and So Was William Rowan Hamilton (their creator)

A Biography and Description of Couplets, Quaternions and the life of Hamilton

Before I get into quaternions and William Rowan Hamilton and why I think both are amazing, I would like to take a couple of minutes to go over why I ended up making this video. This video started because I was working on a detailed history of Maxwell’s equations. That history is why I learned that Maxwell used quaternions in 1873 and then Heaviside and Gibbs (and others) changed his quaternion based equations to vector calculus in the 1880s, but I was struggling to fully understand what was going on as I didn’t really get how quaternions work.

So, as I do, I started to look into the history of the creator of the quaternions: William Rowan Hamilton. I quickly found him to be a fascinating man whose personality and actions have been wildly misunderstood by most historians. It was also in this research that I found a letter from Hamilton describing how his quaternions were developed as an extension to what Hamilton called “my previously published Theory of Couplets.”[1] I immediately looked into this “couplets” or “couple” theory of complex numbers and, suddenly everything clicked.

I then felt like I had two interesting stories to tell, a biography of Hamilton and why he is so misunderstood as well as a description of the math of couples/couplets leading to the math of quaternions and vector calculus.  So I decided to try something new, I have decided to alternate between a biography of William Rowan Hamilton with a description of the real mathematics in 5 parts so if you just want the history, you can skip the math and if you just want the math, you can skip the history and if, like me, you want both you can enjoy both. Ready? Let’s Go!

Part 1: Hamilton’s Professional Life (History)

Almost as soon as William Rowan Hamilton was born to a middle-class Protestant family in Dublin, Ireland in 1805, no one could get over how smart he was. When Hamilton was just 4 years old his mother gushed that she found her son to be, “one of the most surprising children you can imagine; it is scarcely credible…his reciting is astonishing, and his clear and accurate knowledge of geography is beyond belief… but you will think this nothing when I tell you that he reads Latin, Greek, and Hebrew!!” Yes, you heard that right, Hamilton could read 4 languages by the time he was 4! 

By 1815, when William was 9, his father boasted to a friend that, “William continues his even course of commanding and persevering talent” proficient in “Hebrew, Persian, Arabic, Sanskrit, Chaldee, Syriac, Hindoostanee, Malay, Mahratta, Bengali” and was about to start on Chinese![2]

Tragically, when Hamilton was 11 his mother died and 2 years later his father died as well, , leaving 14-year-old Hamilton in the care of his tutor and Uncle James (Hamilton’s 4 sisters were cared for by another Aunt and Uncle).[3] When Hamilton was 16, his Uncle gave him a book on Analytic geometry[4] and he fell into a deep and lifelong love of mathematical functions and geometry which were to dominate his life from then on. Although his Uncle James continued to push the classics, he told another uncle, “I fear I shall never be so fond of them as of the mathematics that I am now reading… [after all] who would not rather have the fame of Archimedes than that of his conqueror Marcellus?”[5] By the next year, a friend showed some of Hamilton’s mathematical work to the Irish Royal Astronomer named Dr. John Brinkley who was very impressed, and made Hamilton a bit of a local celebrity.[6] The author Maria Edgeworth noted in a letter that 18-year-old Hamilton was a “real prodigy of talents” and that, “Dr. Brinkley says he may be a second Newton.”[7]

In college, Hamilton excelled winning first in every subject. They have this award called ‘optime’ or ultimate that is very rarely given to anyone and considered a lifelong honor. Hamilton won it twice![8]

Then, before Hamilton had even graduated college, Hamilton’s mentor, John Brinkley was made into the Bishop of Cloyne and the now 21-year-old Hamilton became the new Royal Astronomer of Ireland. Now ensconced at Dunsink Observatory outside of Dublin, Hamilton setup house with 2 of his sisters, Eliza and Grace and held astronomy lectures which, by the 1830s became very popular with poets and writers, especially as both William and his sister Eliza were published poets and friends with William Wordsworth.[9]

As well as poetry and astronomy, Hamilton also published a lot on the mathematics of science. For example, in April of 1834, Hamilton published a paper on a new method of dealing with the math  of dynamical systems by creating an operator to represent the total energy of the system.[10] These turned out to be very useful once quantum mechanics was developed and are now called Hamiltonians in his honor.[11]  At the time, Hamilton was even more famous for his work on geometric optics and by August, 1835, Hamilton’s reputation was so great that the Viceroy of Ireland decided to honor the now 29-year-old Hamilton with a knighthood, becoming, according to an eye-witness the first person to be “knighted by a Lord Lieutenant either for scientific or literary merit.”[12]

He remained Ireland’s most famous mathematician and scientist for the rest of his life. Now let’s go into some specifics of some of Hamilton’s mathematics. I want to start a year before he was knighted in September, 1834 when Hamilton created his “Theory of Couples/Couplets” to represent complex numbers which, as I said in the introduction, I found helpful to understand quaternions. 

Part 2: The Theory of Couples (Math)

Let me backtrack a bit to explain what real, imaginary and complex numbers are.  Imagine you are solving the problem x^2 = 9 . Then clearly x = 3 is a solution (as 3^2=9 ). But also, as (-1)^2 = 1 , x = -3 is also a solution. So x^2 = 9 has two real solutions, x = \pm 3 .

Now, imagine you are solving the problem x^2 = -9 ., in that case you could say that there is no real answer, which is correct. However, back in 1637, the French philosopher and mathematician René Descartes decided that there were solutions that were, in his words, “imaginary.”[13] This idea was then taken up by Euler in the 1740s where he defined the term imaginary to be equal to the square root of negative 1, and, in 1751 noted that if you have imaginary solutions, they always come in pairs.[14]

Therefore, in the case of x^2 = -9 , x has two imaginary solutions, x = 3i . or x = -3i , where i , the imaginary number is equal to \sqrt{-1}.

But what if you had a problem with a complicated solution, like (y-1)^2 = -9? In that case, y-1=3i or y-1=-3i . If you add 1 to both sides y = 1+3i  or y = 1-3i . In 1829 this was logically called a complex number by Carl Friedrich Gauss as it both has a real number (1) and an imaginary number 3i  (Gauss also popularized using i as the imaginary number \sqrt{-1}).[15]

Five years after Gauss, on September 6, 1834, Hamilton gave a talk called the “Theory of Couples” where he represented complex numbers not as a+ib but as a pair of real numbers (a,b) which you can think of as a pair of numbers are a scale where the x-axis is real numbers and the y-direction is the imaginary numbers. He also made rules for addition and multiplication for these couplets, which would enable, “us to generalize any ordinary algebraic equation from single quantities to pairs, and so to interpret the research of all its roots without introducing imaginaries”[16] 


Let me explain this in a little more depth. Imagine you have two complex numbers and you want to multiply them. You can just multiply it out. So:

But since i^2=-1 , you get


Or, you can combine terms:


So that the real components are xa-yb and the imaginary ones are xb+ya . That is why in Hamilton’s couple notation, he wrote:


This method continues to be used regularly to this day. Now let’s get back to Hamilton’s life.

Part 3: Hamilton’s Love Life (History)

Back on August 17, 1824, when Hamilton was only 19-years-old he met a beautiful 24-year-old woman named Catherine Disney and quickly became smitten.[17] By January, 1825 Hamilton was writing to his Uncle James that “Miss. Disney, beautiful as she is, [is appealing but it is] her mind and her heart, with those who know her, [that] are the objects which engage their attention and secure their love.”[18] Then, tragically for the couple in February, 1825, Hamilton was told by Disney’s mother that his love was going to marry another man.[19] Poor Hamilton was so overwrought that for a brief moment he contemplated jumping into a river and killing himself. Years later he recalled that he couldn’t commit suicide because “I would not leave my post; I felt that I had something to do.”[20] (Decades later Hamilton learned that Catherine was forced to marry and wanted to be with him, but by then it was far too late). 

Burnt, Hamilton focused on his work, although he always felt that the only method to personal and religious happiness was through marriage. It was a full six years later, 1831 that Hamilton decided that he had fully moved on and formed an attachment to an astronomy lover named Ellen de Vere, whose parents encouraged the match. However, after she realized his attachment, she gently rebuffed him by declaring that she couldn’t happily live away from home.[21]  Hamilton was deeply saddened by her declaration but didn’t get mad or bitter and instead declared that, “having had an attachment to a worthy object, and having met with a return of friendship…is not to be regretted, whatever grief it may occasion.”[22]

By the fall of the following year, Hamilton turned his affection to a woman named Helen Bayly “whom I have long known and respected and liked.”[23] Hamilton was truly smitten, and this time both the lady and her family were for the match. By November he wrote to Helen’s sister that he needed to be more careful as he accidentally called the “Biela comet”, “Bayly’s” comet after her.[24] Then a week later he heard that both Helen and her elderly mother were ill, and wrote a loving letter declaring, “How gladly would I, if I were permitted, minister by your sick bed and try to soothe and comfort you.” [25] Helen recovered quickly without Hamilton’s help and by December, 1832 she had formally agreed to the match. By April 1833, William Rowan Hamilton and Helen Bayly got married. 

Despite what you might have heard, they had a deeply loving marriage. For example, in June 1833 when Hamilton learned he had to wait over the weekend to return to his bride after a meeting, sent her a letter of love and a poem that ended with how, “the vow that gave me Helen gave me peace and balm.”[26]

In return, Helen Hamilton wrote her mother that her husband’s “whole happiness seems to lie in making others happy, indeed any woman is blessed to be married to such an affectionate kind creature as Hamilton.”[27]

By May 10, 1834, Helen gave birth to their first child, named William Edwin and a little over a year later, on August 4, 1835 a boy named Archibald. However, by the time she had her third child in 1840 (to a girl named Helen Eliza), things went downhill with Helen’s health. (Helen was diagnosed with a “nervous illness” whatever that means).[28] Hamilton wrote to a friend, “My anxiety about Lady Hamilton’s health has made me very unfit for writing for many months past.”[29] For the next year and a half, Helen was sent for months to recuperate with her sisters, and the young family struggled without her and were helped by Hamilton’s sister Sydney.[30]  Helen eventually returned home in January 1842 to everyone’s delight (although she had one other 6 month “spell” in 1856).[31] Despite her health issues, they remained devoted to each other all of their lives. As Hamilton’s biographer Robert Graves, who knew them personally, wrote in 1885, “[Hamilton] remained to the end of his life an attached husband, as Lady Hamilton remained an attached wife, as well as a good woman.”[32] Anyway, it was on October 16, 1843, almost 2 years after her return to home that Helen accompanied her husband on a walk to a meeting. This walk was special because it was the walk when William Hamilton had an epiphany about quaternions. Which brings me, finally, to what quaternions are and how they work.

Part 4: Quaternions! (Math)

After his paper of “couples” or “couplets” in 1834, Hamilton started to think about expanding that theory to a new theory of “triplets” or something with a real axis and 2 imaginary axes: i and j . As he wrote his friend Robert Graves, “Since \sqrt{-1}  is in a certain well-known sense, a line perpendicular to the line 1, it seemed natural that there should be some other imaginary to express a line perpendicular to both the former… Calling the old root, as the Germans often do i , and the new one j I inquired what laws out to be assumed for multiplying together  a+ib+jc and x+iy+jz .”

However, Hamilton was stumped on how to multiply them, writing, “but what are we to do with ij?”[33] He tried making ij  equal to 1 or to -1, but neither of those choices worked so that the sum of the squares of the coefficients in the product equaled the sum of the squares of the factors. In fact, in order to make the multiplication work out ij had to equal -ji ! Now of course he could have just defined ij=-ji=0, which would work, but Hamilton found it to be “odd and uncomfortable” to delete possibly such an important term. Therefore, he defined a new letter k, where k=ij and k=-ji, with the idea that he would somehow prove whether k had to be zero or not.[34] 

I would like to take a moment to acknowledge what an astonishing and revolutionary act this was. Hamilton was working at least 7 years before the invention of matrices where AB \neq BA.[35] In fact, before Hamilton there were no formal mathematical operations where the order of operations mattered, nowhere else where ij \neq ji . But Hamilton just decided that it was possible, but he couldn’t figure out how to make it work in a systematic way and by September of 1843 it became a bit of an obsession.

According to Hamilton, his sons, who were just 8 and 9 years old, would ask him every morning for a month “Well, Papa, can you multiply triples?” And he was forced to reply, as he put it, “with a sad shake of the head: ‘No, I can only add and subtract them’.”[36]

But on that Monday, as he and his wife were walking along a canal to a meeting, it just came to him. Not only was his variable k not equal to zero, he could demonstrate how, like i^2 and j^2, k^2 also equaled -1. Let me show you. Since k=ij=-ji , he could define:


But since j^2=-1, then -j^2=-1, so 


 Hamilton then realized that would all make sense if k was a third imaginary axis. His system would therefore have 4 parts, a real part and three imaginary dimensions i, j and k :


The fact that it has 4 parts is why Hamilton instantly called it a quaternion (quarter for 4). As he wrote his friend the next day, “we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples.”[37] 

With this setup, Hamilton needed a way to multiply k times j and k times i. However, Hamilton saw instantly that he could use the definition of k with the knowledge that i^2=j^2=-1 , to solve for all the other multiplication rules.

As Hamilton put it, “I saw the we had probably ik=-j, because ik=i(ij) , and i^2=-1 .” [38] In reverse if you multiply k times i, (and use k=-ji) you get ki=(-ji)i=j(-i^2)=j, so that in symmetry with the definition of k which equals both ij and -ji j=ki=-ik.

Similarly, jk=j(-ji)=-j^2i=i . And kj=(ij)j=ij^2=-i so i=jk=-kj

In total Hamilton’s quaternion rules are:


And the set of equations

i=jk=-kj , j=ki=-ik, and k=ij=-ji

Hamilton then simplified these equations to a single statement:


(Note that all 6 equations relating i, j and k can be found multiplying ijk=-1 by different letters and using the fact that any of the letters squared is negative 1, you should try it out, it is pretty cool).

Hamilton was overwhelmed with excitement with his discovery, which all took place in the matter of minutes while he was walking near a bridge and, as he told his son Archibald, “I [could not] resist the impulse… to cut with a knife on a stone of [the] Bridge, as we passed it, the fundamental formula with the symbols i, j, k , namely,

 i^2=j^2=k^2=ijk=-1 [39]

[Unfortunately, Hamilton’s graffiti has faded over time and been replaced by a plaque]. 

Now that Hamilton had these relations, he could make rules to multiply 4-part quaternions just as he had rules to multiply 2-part couplets.[40] By July, 1846, Hamilton added new vocabulary to the quaternion. He called the real term the scalar and he called the terms with i,j or k the “vector.”[41] Note that initially Hamilton wasn’t thinking of the vector as we do now, as something with values in the x, y, or z directions, but merely as a way of distinguishing the i, j and k terms from the scalar terms. However, you could take a vector, W, and represent it as a quaternion with no scalar component:


And you could multiply it by another vector lower case w as another quaternion with no scalar component:


You will get a result with 9 terms:

Ww=(Xxi^2+Yyj^2+Zzk^2) + (Yzjk+Zykj)+ (Zxki+Xzik)+ (Xyij+Yxji)

Using Hamilton’s rules that i^2=j^2=k^2=-1, and that jk=-kj=i, ki=-ik=j, and ij=-ji=k it changes to:

Ww=-(Xx+Yy+Zz) + (Yz-Zy)i\ + (Zx-Xz)j+(Xy-Yx)k

Which you can think of it having two results, a scalar:

 Scalar Ww=-(Xx+Yy+Zz)

And a more complicated vector result:

Vector Ww=(Yz-Zy)i + (Zx-Xz)j + (Xy-Yx)k

In addition, in October, 1847, Hamilton decided that since physics often used the operation:


It made sense to Hamilton to make an unnamed quaternion operator that was defined as:


Because, in that case the vector result goes to zero and the scalar result is:[42]


(Note that I call this the “del” operator, as that is one of the names eventually given to this operator.) Also notice that because this del squared operator is a scalar, one can use it without quaternions at all, which Maxwell did (without the negative sign) in 1864.[43]

Anyway, Hamilton noted in 1847 that, “perhaps not less remarkable, nor having less extensive consequences”[44] is that if he used his function times any other vector, (t,u,v) you would get a quaternion where the scalar part is:


 and the vector result is:


Hamilton’s explained his logic in more detail in his book, “Lectures in Quaternions,” written in 1853, where he added that this operator, “must yet become extensively useful in the mathematical study of nature” especially with regard to “the modern researches in analytical physics, respecting attraction, heat, electricity, [and] magnetism.”[45]

Hamilton’s statements about electricity and magnetism were particularly prescient as as soon as James Clerk Maxwell read his friend Peter Tait’s book on quaternions written in 1867, Maxwell started to find a lot of uses for it in his magnetoelectric equations and started to add new names to all the functions. For example, he named the Scalar \nabla the convergence of the function as it showed how a function changed as it converged on a point.  In addition, he called the Vector \nabla the “twirl” or perhaps the “curl” of the function as it represented how a function changed while curling or twirling around a point. [46]

These functions were so helpful that in 1873, Maxwell, who had created his electrodynamic equations without vectors, published a book restating his theories of electricity (Maxwell’s equations) using quaternions and introduced the world to the concepts of “convergence” and “curl.”[47] It was that book that inspired a telegraph operator named Oliver Heaviside to quit his job and devote himself to understanding Maxwell’s equations.[48] By December 2, 1882, Heaviside decided that although quaternions are, “a very remarkable system of mathematics,”[49] he found that the operations were too difficult and complicated to use regularly. However, Heaviside decided it would be useful to use “Maxwell’s” concepts of curl and convergence derived from Hamilton’s quaternions without dealing with the other features of Hamilton’s mathematics. I.e. to define the convergence as:

convergence W=-(\frac{dX}{dx}+\frac{dY}{dy}+\frac{dZ}{dz})

and to define the curl separately for each direction as:[50]

curl W_x=(\frac{dZ}{dy}-\frac{dY}{dz}), curl W_y=(\frac{dX}{dz}-\frac{dZ}{dx}), and curl W_z=(\frac{dX}{dz}-\frac{dZ}{dx})

Then, in April of 1883 Heaviside decided to remove the negative term from the convergence and call it a divergence stating: “we may as well use the term ‘divergence’ for the same quantity with the + sign prefixed [so that] if the amount of divergence be positive, it indicates positive electricity.”[51]  Therefore, the divergence of W would be:

divergence \frac{dX}{dx}+\frac{dY}{dy}+\frac{dZ}{dz}

As Heaviside wrote in 1891, “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.”[52] It was only a year after Heaviside defined the divergence when the American physicist Josiah Willard Gibbs put together a little pamphlet for his students titled “Elements of Vector Analysis” that took Heaviside’s methods of using Hamilton’s multiplication rules without his quaternions to the next level. Or as Gibbs put it, “the following analysis [should be] familiar under a slightly different form to students of quaternions” but  “does not require any use of the conception of quaternions, being simply to give a suitable notation for those relations between vectors.”[53]

In this book, Gibbs made a new formal method of mathematics that we use to this day. In this way, Gibbs gave us the power to use the relations that Hamilton conceived in 1843 without having to think in 4-dimensional imaginary space.

In Gibbs work, a vector has the current definition of a vector and the three directions were represented by the directions i, j, and k (which are now unit direction vectors and *not* equal to the square root of negative 1). Gibbs then created two ways to multiply vectors, either with what he called the “direct product” (which we now call the dot product) or what he called the “skew product” (we now call the cross product), which were inspired by the scalar and the vector results of quaternion multiplication.[54] Recall that if you have a quaternion W, and multiply it to another quaternion w, the scalar result is:


Now compare that to Gibbs “direct” or dot product:

\vec{W}\cdot \vec{w}=Xx+Yy+Zz

Which only differs by the removal of the negative sign. In addition, the vector result of the multiplication of the quaternions is a vector:

VectorWw=(Yz-Zy)i + (Zx-Xz)j + (Xy-Yx)k

Which is exactly the same as Gibb’s result for the “skew” or cross product

\vec{W} \times \vec{w}=(Yz-Zy)i + (Zx-Xz)j + (Xy-Yx)k

Gibbs also realized that he could relate Heaviside’s “curl” and “divergence” vectors to his cross and dot product multiplication with Hamilton’s triangle operator so that he wrote that “\nabla \cdot w is called the divergence of w and \nabla \times w its curl [where].[55]

[Divergence of w] =\nabla \cdot w = \frac{dX}{dx}+\frac{dY}{dy}+\frac{dZ}{dz}

[Curl of w =] \nabla \times w= (\frac{dv}{dy}-\frac{du}{dz})i+(\frac{dt}{dz}-\frac{dv}{dx})j+(\frac{du}{dx}-\frac{dt}{dy})k 

Which continues to be the definition of divergence and curl. With this you can clearly see how vector calculus was based on Hamilton’s mathematics. But one can also see how removing the rules of vector mathematics from the quaternions could be advantageous for many situations, especially anything to do with surface integrals. However, Gibbs did not publish his book (although he did send a copy to Heaviside in 1888)[56] so it was mostly ignored until Gibb’s student Edwin Bidwell Wilson published a book of “Vector Analysis” in 1901 which was a “text-book …founded upon the lectures of J Wiliard Gibbs.”[57] Therefore, it was only a few years before Gibbs’s death that the scientific and mathematical community started to use Gibbs’ methods of vector calculus as we continue to do to this day.  

Unfortunately, William Rowan Hamilton did not live long enough to see how his quaternions were transformed into vector calculus. Still, he was fully convinced that quaternions were amazing and would soon be acknowledged as much. In February 1854 he said that “the quaternions have changed the face of algebra completely” and that “mathematics used to be called French Mathematics when he was a child, but that the world would have to learn Irish Mathematics soon.”[58] Instead, Hamilton was either forgotten or only known as a drunk who hated his wife. Which brings me to…

Part 5: Hamilton’s Legacy (History)

Hamilton felt that his quaternions were the most important thing he had discovered, and therefore focused the majority of his research on them for the rest of his life. Tragically, by the 1860s, Hamilton’s health started to falter due to gout and overwork. By September 2, 1865 after a bout of bronchitis he died peacefully just after his 60th birthday. 

After Hamilton died a clergyman named Robert Graves (who was the brother of Hamilton’s collaborator and friend John Graves) took 24 years to put together three books with over 2,000 pages worth of personal letters to, from, and about Hamilton. Interspersed in these letters is a story about how in February 1846, at a Geological Society Meeting, Hamilton got overheated about a scientific debate. Hamilton wrote in a private letter that “the excitement of the conversation, the speeches, and the wine… turned out to be more than I could bear… and I was seized with a giddiness and rush of blood to the head which totally incapacitated me from…keeping my ideas under my control.”[59]

Robert Graves wrote that after this event his other brother Reverend Charles Graves visited Hamilton and suggested that due to the effect on his reputation, Hamilton should “adopt a regimen of entire abstinence from alcoholic stimulants,” which, for a while, he did.[60] 

However, two years later, in 1848, Hamilton had a glass of champagne at a party and began drinking again occasionally at dinner parties. The author’s brother, Charles, once again visited Hamilton, but this time although Hamilton “gratefully received” his friend, Hamilton decided that temperance could be defined as restraint not abstinence, which is how he acted from then on.[61]

Many years after Hamilton’s death Robert Graves tried to make sense of how his hero could reject his brother’s council. Graves decided that it was all, basically, due to Helen Hamilton’s health issues. As I noted earlier Graves wrote that William and Helen Hamilton were devoted to each other for their entire lives, but Graves found that unfortunate.

Graves felt that Hamilton needed governing to keep him from what Graves felt were his worse instincts of overwork and occasional drink and decided it was Helen Hamilton’s fault for not being what he called “capable.” Specifically, Graves made up this whole narrative that when Helen returned after her attack of “anxiety” in 1842, she wasn’t as good at managing their staff of servants as Graves felt she should be. It was this, Graves declared, that caused Hamilton to be without a fire or hot chocolate late at night, and forced him to sip porter to keep warm, porter that was “fraught with inevitable danger.”[62]

Note that almost every high-ranking aristocrat drank at dinner parties (Hamilton’s successor once quipped that he was going to pretend to be a teetotaler just to get away from the relentless amount of liquor provided). However, to Graves somehow Helen Hamilton’s unfortunate health issues forced her poor husband to have an occasional night cap, and then, according to Graves “the insidious habit gradually gained firmer possession, and produced that relentless craving which in a few years from this time exercised over him an occasional mastery.”[63] You can see how people got the impression from these overwrought words of Graves that Hamilton was a drunk and that was due to a failing of his wife.

Then, in 1937 a mathematician named Eric Bell took the story to the next level in a short biography titled “An Irish Tragedy.” In this version, Bell weaved a story where Hamilton only married Helen out of pity and loneliness. Hamilton was “properly hooked by an ailing female… who either through incompetence or ill-health, let her husband’s slovenly servants run his house as they chose.” Unlike Graves, Bell had nothing but contempt for Helen calling her a “weakling” whose behavior forced Hamilton to take “nourishment from a bottle.”[64]

Anyway, as Bell’s 20-page diatribe was far easier to digest than Graves’s 2000 page detailed and mostly measured series of books, Bell’s version of events became very influential. However, it seemed odd that a man would end up a drunk due to mismanaged servants, and historians started to look for other clues. That was when some people realized that Hamilton’s first love had been beautiful and forced to marry another man against her will. Due to this, the story changed to one where William Hamilton was pining for Catherine Disney his entire life and therefore never loved his wife, Helen, and it was this lovesick story that caused him to turn to drink. That is the story that is prevalent today.[65]

My favorite example is a truly delightful acapella song about “William Rowan Hamilton” to the tune of “Alexander Hamilton” let me show you a bit and you can see what I mean:[66]

That version of history only started to change recently, just a year after the people at “acapella science” published that video, in 2017, when a Dutch researcher named Anne van Weerden published a book defending the reputation of both William and Helen Hamilton titled “A Victorian Marriage.”[67] Due to her years of dedicated research people are starting to see how they might have been perpetuating Victorian views and not seeing the big picture,[68] and I hope that this video will help her quest to correct this historical wrong.

There is one aspect of Hamilton’s reputation that has been besmirched almost as much as his personal life, and that is the reputation of the quaternions. For although quaternions are highly respected in mathematical and computer programing circles, many physicists are under the false impression that quaternions were a mistake and unrelated to vector calculus. As the author Eric Bell (the guy who thought Helen Hamilton was a disgusting “weakling” who caused Hamilton’s drunken and miserable state) declared in 1937, “Hamilton’s deepest tragedy was neither alcohol nor marriage but his obstinate belief that quaternions held the key to the mathematics of the physical universe.”[69]

The strange thing is that Bell should have known that Hamilton was right, quaternions are the key to the mathematics of the physical universe. Why do so many people get taught that quaternions were replaced with vector calculus instead of the inspiration for vector calculus? Well, it has to do with a pretty vicious debate between Oliver Heaviside and Hamilton’s protegee Peter Tait (with input from Maxwell, Gibbs, Heinrich Hertz and more). And that story is next time on: “The Evolution of Wireless.”

Thanks for watching my video. Some of you might have noticed that I changed my tagline from “The Lightning Tamers” to “The Evolution of Wireless” and that is because I have finished my videos about how electricity was tamed and put into our homes and have published it as my first book also titled “The Lightning Tamers.” I am now working on the videos on my next series “The Evolution of Wireless: from Sir Isaac Newton to Hedy Lamar” which will be my third book as well as working on my second book “The Radium Revolution,” so click that subscribe and hit that bell because I have a lot of great stories to tell. Also, I put the script for this with clickable links on my website www.KathyLovesPhysics.com. Big thank you to my patrons I have a link to join their ranks below and a special thanks to Anne van Weerden not only for her incredible and dedicated research but for personally taking so much of her precious time to help a stranger get this story right. I also put a link to her book/website below, with a lot more detail and information. Stay safe and curious. Bye.  

[1] Hamilton to John Graves (Oct 17, 1843) Brewster The London and Edinburgh Phil Magazine vol 25 (1844) p. 491 https://www.g oogle.com/books/edition/The_London_and_Edinburgh_Philosophical_M/5TxZAAAAcAAJ?hl=en&gbpv=1

[2] Archibald Hamilton to Mr. Beilby (May 18, 1815) found in Graves, Robert Life of Sir William Rowan Hamilton (1882) p. 45 https://archive.org/details/lifeofsirwilliam01gravuoft/page/45/mode/2up

[3] Weerden, Anne van A Victorian Marriage (2017) p. 27 & 487 https://www.google.com/books/edition/A_Victorian_Marriage/6Ow9DwAAQBAJ?hl=en&gbpv=0

[4] Hamilton to Cousin Arthur (Sept 4, 1822) Graves, Robert Life of Sir William Hamilton (1882) p. 112 https://archive.org/details/lifeofsirwilliam01gravuoft/page/112/mode/2up

Analytic Geometry (1819) https://digitalcollections.tcd.ie/concern/works/qj72pf99m

[5] Hamilton to Mary Hutton (Aug 26, 1822) found in Graves, Robert Life of Sir William Rowan Hamilton (1882) p. 110 https://archive.org/details/lifeofsirwilliam01gravuoft/page/110/mode/2up

[6] William Hamilton to Arthur Hamilton (Oct, 1822) found in Graves, Robert Life of Sir William Hamilton (1882) p. 119 https://archive.org/details/lifeofsirwilliam01gravuoft/page/119/mode/2up

[7] Maria Edgeworth to Honora Edgeworth (Aug 28, 1824) The Life and Letters of Maria Edgeworth vol. 2 (1895) p. 473

The Life and Letters of Maria Edgeworth – Google Books

[8] Hamilton to Sister Eliza (Nov 7, 1826) Graves, Robert Life of Sir William Rowan Hamilton (1882) p. 221 https://archive.org/details/lifeofsirwilliam01gravuoft/page/221/mode/2up

[9] Hamilton to Eliza Hamilton (Sept 23, 1829) Graves, Robert Life of Sir William Rowan Hamilton (1882) p. 342 https://archive.org/details/lifeofsirwilliam01gravuoft/page/342/mode/2up  You can read some of Eliza Hamilton’s poetry here: https://www.google.com/books/edition/Poems/8DdkAAAAcAAJ?hl=en&gbpv=1

[10] Hamilton “On a General Method in Dynamics” (April 10, 1834) Phil Trans of the Royal Society (part 2 for 1834) p. 247 https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/GenMeth.pdf

[11] For example, Scott, Brice Elements of Quantum Optics (2019) p. 98-9 https://www.google.com/books/edition/Elements_of_Quantum_Optics/G-TEDwAAQBAJ?hl=en&gbpv=0

[12] Tichnor, Life, Letters, and Journals of George Tichnor vol. 1 (1876) p. 351 https://www.google.com/books/edition/Life_Letters_and_Journals_of_George_Tich/AbIR–mlvh0C?hl=en&gbpv=1

[13] The Geometry of René Descartes (translated in 1925) p. 175 https://www.google.com/books/edition/The_Geometry_of_René_Descartes/2UbQAAAAMAAJ?hl=en&gbpv=0

[14] Euler “Investigations on the Imaginary Roots of Equations” (1749) Mémoires de l’académie des sciences de Berlin vol. 5, 1751, pp. 222–288 translated by Todd Doucet Euler_170_Doucet.pdf (maa.org) or in original “Reserches sur les racines imaginaires des equations” p. 225 E170.pdf (maa.org)

[15]Gauss “Principia Generalia Theoriae Figurae Fluidorum in Statu Aequilibrii” (Sept 28, 1829) Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). vol. 7 pg. 97 #293 – Commentationes Societatis Regiae Scientiarum … v.7 1828-1831. – Full View | HathiTrust Digital Library

[16] Hamilton “On Conjugate Functions, or Algebraic Couples” (Sept, 1834) Report of the British Association for the Advancement of Science, Fourth Meeting (1835) p. 519 https://www.google.com/books/edition/Fourth_Meeting_Held_At_Edinburgh_In_1834/P79ZAAAAcAAJ?hl=en&gbpv=1

[17] Graves The Life of Sir William rowan Hamilton, vol. 1 (1882) p. 160 https://archive.org/details/lifeofsirwilliam01gravuoft/page/160/mode/2up

[18] Hamilton to Uncle James (Jan 11, 1835) Graves The Life of Sir William rowan Hamilton, vol. 1 (1882) p. 173 https://archive.org/details/lifeofsirwilliam01gravuoft/page/173/mode/2up

[19] Graves The Life of Sir William rowan Hamilton, vol. 1 (1882) p. 182 https://archive.org/details/lifeofsirwilliam01gravuoft/page/182/mode/2up

[20] Hamilton (October 6, 1858) Graves, The Life of Sir William Rowan Hamilton, vol 2 (1882) p. 610 https://archive.org/details/lifeofsirwilliam02gravuoft/page/610/mode/2up

[21] Graves The Life of Sir William rowan Hamilton, vol. 1 (1882) p. 505 https://archive.org/details/lifeofsirwilliam01gravuoft/page/505/mode/2up

[22] William Hamilton to sister Eliza (Dec 19, 1831) Graves The Life of Sir William rowan Hamilton, vol. 1 (1882) p. 508 https://archive.org/details/lifeofsirwilliam01gravuoft/page/508/mode/2up

[23] William Hamilton to Aubrey de Vere (Nov 7, 1832) Graves The Life of Sir William rowan Hamilton, vol. 1 (1882) p. 622 https://archive.org/details/lifeofsirwilliam01gravuoft/page/622/mode/2up

[24] Hamilton to Mrs. W Rathborne (Nov 22, 1832) Graves, The Life of Sir William Rowan Hamilton, vol 2 (1882) p. 11 https://archive.org/details/lifeofsirwilliam02gravuoft/page/11/mode/2up

[25] Hamilton to Helen Bayly (Nov 26, 1832) Graves, The Life of Sir William Rowan Hamilton, vol 2 (1882) p. 11-2 https://archive.org/details/lifeofsirwilliam02gravuoft/page/11/mode/2up

[26] Hamilton to Helen Hamilton (June 29, 1833) Graves, The Life of Sir William Rowan Hamilton, vol 2 (1882) p. 50 https://archive.org/details/lifeofsirwilliam02gravuoft/page/50/mode/2up

[27] Helen Hamilton to Mrs. Bayly (Oct 5, 1833) Graves, The Life of Sir William Rowan Hamilton, vol 2 (1882) p. 61 https://archive.org/details/lifeofsirwilliam02gravuoft/page/61/mode/2up

[28] Graves, The Life of Sir William Rowan Hamilton, vol 3 (1889) p. 51


[29] Hamilton to Viscount Adare (Sept 14, 1840) Graves, The Life of Sir William Rowan Hamilton, vol 2 (1882) p. 324 https://archive.org/details/lifeofsirwilliam02gravuoft/page/324/mode/2up

[30] Graves, The Life of Sir William Rowan Hamilton, vol 2 (1885) p. 334 https://archive.org/details/lifeofsirwilliam02gravuoft/page/334/mode/2up

[31] Graves, The Life of Sir William Rowan Hamilton, vol 3 (1889) p. 51


[32] Graves, The Life of Sir William Rowan Hamilton, vol 2 (1885) p. 335 https://archive.org/details/lifeofsirwilliam02gravuoft/page/335/mode/2up

[33] Hamilton to John Graves (Oct 17, 1843) Brewster The London and Edinburgh Phil Magazine vol 25 (1844) p. 490 https://www.google.com/books/edition/The_London_and_Edinburgh_Philosophical_M/5TxZAAAAcAAJ?hl=en&gbpv=1

[34] Hamilton to John Graves (Oct 17, 1843) Brewster The London and Edinburgh Phil Magazine vol 25 (1844) p. 491 https://www.g oogle.com/books/edition/The_London_and_Edinburgh_Philosophical_M/5TxZAAAAcAAJ?hl=en&gbpv=1

[35] J J Sylvester “Additions to ‘On a new Class of Theorems” The London, Edinburgh and Dublin Phil. Magazine (1850) vol. 37 p. 369 The London, Edinburgh and Dublin Philosophical Magazine and Journal of Scie… – Google Books

[36] Hamilton to Archibald Hamilton (Aug 5, 1865) Life of Sir William Rowan Hamilton vol 2 (1885) p. 434 https://archive.org/details/lifeofsirwilliam02gravuoft/page/434/mode/2up

[37] Hamilton to John Graves (Oct 17, 1843) Brewster The London and Edinburgh Phil Magazine vol 25 (1844) p. 491 https://www.google.com/books/edition/The_London_and_Edinburgh_Philosophical_M/5TxZAAAAcAAJ?hl=en&gbpv=1

[38] Hamilton to John Graves (Oct 17, 1843) The London and Edinburgh Phil Magazine vol 25 (1844) p. 492 https://www.google.com/books/edition/The_London_and_Edinburgh_Philosophical_M/5TxZAAAAcAAJ?hl=en&gbpv=1

[39] Hamilton to Archibald Hamilton (Aug 5, 1865) Life of Sir William Rowan Hamilton vol 2 (1885) p. 435 https://archive.org/details/lifeofsirwilliam02gravuoft/page/435/mode/2up

[40] Hamilton “On a new Species of Imaginary Quantities Connected with a Theory of Quaternions” Proceedings of the Royal Irish Academy, vol. 2, (1844) p. 424 https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern1/Quatern1.pdf

[41] 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

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

 The London, Edinburgh and Dublin Philosophical Magazine and Journal of Scie… – Google Books

[43]Maxwell “A Dynamical Theory of the Electromagnetic Field” (1864) The Scientific Papers of James Clerk Maxwell (1890) p. 578 The Scientific Papers of James Clerk Maxwell … – Google Books

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

 The London, Edinburgh and Dublin Philosophical Magazine and Journal of Scie… – Google Books

[45] (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 

[46] Maxwell to Tait (Nov 7, 1870) found in Knott, Cargill Life and Scientific Work of Peter Guthrie Tait (2015) p. 143-4 https://www.google.com/books/edition/Life_and_Scientific_Work_of_Peter_Guthri/LP0GBwAAQBAJ?hl=en&gbpv=1

[47] Maxwell, James Clerk A Treatise on Electricity and Magnetism: vol. 1 (1873) p. 19 & 28 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] Paul Nehin Oliver Heaviside (2002) p. 24-25 https://www.google.com/books/edition/Oliver_Heaviside/e9wEntQmA0IC?hl=en&gbpv=0

[49] 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

[50] For example, he defined curl of the magnetic field in Heaviside, Oliver “Magnetic Force and Current -III” (Jan 6, 1883) The Electricians vol. 10 p. 175 https://www.google.com/books/edition/_/qmlNAAAAYAAJ?gbpv=0

[51] 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

[52] 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)

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

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

[55] 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

[56] 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)

[57] E. B. Wilson, Vector Analysis (1901) Vector analysis, a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs : Gibbs, J. Willard (1839-1903) : Free Download, Borrow, and Streaming : Internet Archive

[58] Hamilton paraphrased in Graves William Rowan Hamilton vol. 2 (1889) p. 693 https://archive.org/details/lifeofsirwilliam02gravuoft/page/692/mode/2up

[59] Graves Life of Sir William Rowan Hamilton vol 2 (1885) p. 509 https://archive.org/details/lifeofsirwilliam02gravuoft/page/509/mode/2up

[60] Graves Life of Sir William Rowan Hamilton vol 2 (1885) p. 507 https://archive.org/details/lifeofsirwilliam02gravuoft/page/507/mode/2up

[61] Graves Life of Sir William Rowan Hamilton vol 2 (1885) p. 632 https://archive.org/details/lifeofsirwilliam02gravuoft/page/632/mode/2up

[62] Graves Life of Sir William Rowan Hamilton vol 2 (1885) p. 332 https://archive.org/details/lifeofsirwilliam02gravuoft/page/332/mode/2up

[63] Graves Life of Sir William Rowan Hamilton vol 2 (1885) p. 507 https://archive.org/details/lifeofsirwilliam02gravuoft/page/507/mode/2up

[64] E. T. Bell Men of Mathematics (1937) p. 387 https://archive.org/details/in.ernet.dli.2015.59359/page/n73/mode/2up

[65] For example, see Kuldeep Singh Linear Algebra: Step by Step (2013) p. 513 Linear Algebra – Google Books

[66] YouTube video song about Hamilton to the tune of “Alexander Hamilton” https://youtu.be/SZXHoWwBcDc

[67] Anne van Weerden A Victorian Marriage (2017) https://www.google.com/books/edition/A_Victorian_Marriage/6Ow9DwAAQBAJ?hl=en&gbpv=0

[68] An example of how Van Weerden changed the narrative can be seen in Gillian O’Neill “Revising the Legacy of William Rowan Hamilton” (Feb 9, 2022) University Times  https://universitytimes.ie/2022/02/revising-the-legacy-of-william-rowan-hamilton/

[69] E. T. Bell Men of Mathematics (1937) p. 396 https://archive.org/details/in.ernet.dli.2015.59359/page/n81/mode/2up?q=obstinate

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