Quaternions are Amazing and so is William Rowan Hamilton!

Transcript

- Before I get into the history of quaternions and how they work and the history of William Rowan Hamilton, and I guess how he worked, I wanted to take a moment to say why I ended up making this video in the first place. This video started because I

was making a series of videos on the detailed history of Maxwell's equations. That is how I ended up learning about how Maxwell wrote his equations in quaternion form in 1873 and how that inspired Heaviside and Gibbs to make vector physics.

But what I was missing was how quaternions work? So as I do, I looked into the history of the creator of quaternions, William Rowan Hamilton, and I found him to be a fascinating person whose biography is wildly misunderstood by most historians. It was also in this research that I found a letter from Hamilton the day after he discovered quaternions, describing how his

quaternions were developed as an extension of what Hamilton called "my published Theory of Couplets." I immediately looked into the history of this couplets or Couple Theory, and suddenly everything clicked. I then felt like I had two interesting stories to tell.

One was a biography of Hamilton and why he's so misunderstood, and one was the history of the development of the math of both couples or couplets, and how that led to triplets and how that led to quaternions and then finally how quaternions work and a little bit about how they developed into vector mathematics. So I decide to do something slightly different in this video. It has five sections, and I'm gonna alternate between

the history and the math. So those of you who just want the history can skip the math, and those of you who just want the math can skip the history and those of you like me, who like both, can watch both.

Ready? Let's go. Part one, Hamilton's professional life. 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. By the time he was four 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 four languages by the time he was four.

By 1815, when he was nine years old, his father noted to a friend that, "William continues his even course of commanding and persevering talent," Proficient in Hebrew, Persian, Arabic, Sanskrit, Chaldee, Syriac, Hindustani, Malay, Mahratta, Bengali, and was about to start on Chinese. Tragically, when Hamilton was 11, his mother died and then a little over two years later, his father died as well.

Leaving 14 year old, Hamilton, in the care of his uncle James, while his four sisters were cared by another pair of aunt and uncles. Hamilton always liked math, but it really took off when he was 16 and his uncle gave him a book of analytic geometry 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 wrote, "I fear I shall never be so fond of them as of the mathematics that I'm now reading." After all, "Who would not rather have the fame of Archimedes than that of his conqueror Marcellus." The next year, a friend showed some of his mathematical work to the Irish royal astronomer

named Dr. John Brinkley, who was very impressed and made Hamilton a bit of a local celebrity. The author, Maria Edgeworth, noted in a letter that the 18 year old Hamilton

was a real prodigy of talents and the Dr. Brinkley says he may be a second Newton. In College Hamilton excelled, getting first in every subject. They have this award called

the Optime, or Ultimate, that is very rarely given to anyone and considered a lifelong honor. Hamilton won it twice. Then, before Hamilton even graduated college, his 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 Dublin, Hamilton set up house with two 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. As well as poetry and astronomy, Hamilton published a lot on the mathematics of physics. For example, in April, 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.

These operators ended up being very useful once quantum mechanics was developed and are now called Hamiltonians in his honor. At the time, Hamilton was even more famous for his work on geometric optics, which arguably led to to the Hamiltonians, and his fame was so great that by August 1835, the Vice Roy of Ireland decided to honor, the now 29 year old, Hamilton with a knighthood, becoming, according to an eyewitness, the first person to be

knighted by a Lord Lieutenant either for scientific or literary merit. He remained Ireland's most famous mathematical scientist for the rest of his life. Now I wanna go into some details of some of his mathematics.

Specifically, I want to go to about a year before he was knighted in September, 1834, when he came up with the theories of couples/couplets to explain complex numbers, which, at least for me, help me understand quaternions. Part two, The Theory of Couples or couplets.

Let me backtrack a bit and explain a tiny bit what real, imaginary and complex numbers are. Say you have a problem, X squared = 9, then clearly X = 3 is a solution, but also as -1 X -1 is 1, x = -3 is also a solution. Now imagine you're solving

the problem X squared = -9. In that case, you can say there is no real answer, which is correct. However, back in 1637, the mathematician and philosopher, Descartes, decided that there were

solutions that were, in his words, imaginary. This idea was then taken up by Euler, where he defined the term imaginary to be equal to the square root of -1. And in 1751, noted that if you have imaginary solutions, they always come in pair. Therefore, in the case of x squared = -9, x has two imaginary solutions, x = + or - 3 times the square root of -1, or x + or - 3i, where

i, the imaginary number, is equal to the square root of -1. But what if you had a problem with a more complicated solution? Like y - 1 squared = -9. In that case, y - 1 = 3i or y - 1 = -3i. And if you add one to both sides, you get y = 1 + 3i or y = 1 - 3i. In 1829, this was logically

called a complex number by the German mathematician, Carl Friedrich Gauss, as it both has a real number, 1, and an imaginary number, + or - 3i. Note that Gauss also popularized using i as the imaginary number, square root of negative 1. Five years later, on September 6th, 1834, Hamilton gave a talk on what he called the Theory of Couples, where

he represented complex numbers not as a + iv or u + iv, but as a pair of real numbers, (a, b). which you can think of a pair of numbers on the scale where the x-axis is the real number and the y-axis is the imaginary numbers. He also made rules for addition, subtraction, multiplication of these complex numbers, which would enable one

to generate any ordinary algebraic equation from a single quantities to pairs. And so interpret the research of all its roots without introducing imaginaries. So I wanna talk about the multiplication in a little bit more depth.

Imagine you have two complex numbers. Well, you can just multiply it out. So x + iy times a + ib will give you four terms for multiplying all four things to each other.

One of the terms has no imaginary term and one of them has i squared, but since i squared = -1, you get two real terms, xa - yb, and two imaginary terms, xb + ya. That is why in Hamilton's couple notation he wrote, (x, y)(a, b) is (xa - yb, xb + ya). This method continues to be used regularly in mathematics to this day. Now let's take a break from the math and go back to Hamilton's life.

Part three, Hamilton's personal life. On August 17th, 1824, when William Hamilton was only a 19 year old college student, he met a beautiful 24 year old woman named Catherine Disney and quickly became smitten. By January, 1825, Hamilton was writing his uncle James that, "Ms.

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." Then tragically for the couple, in February, 1825, Hamilton was told by Disney's mother that she was going to wed another.

Hamilton was distraught and for a brief moment, he even contemplated jumping into a river and killing himself. Years later, he recalled he couldn't commit suicide because "I would not leave my post. I felt I had something to do." Decades later, Hamilton

learned that his love was forced to marry another, but by then it was far too late. Burnt, Hamilton decided to focus on his mathematics and his poetry. Although he always felt that the secret to both personal and religious happiness was marriage.

Then six years later in 1831, Hamilton decided he'd moved fully on and formed an attachment to astronomy lover named Ellen De Vere, whose parents encouraged the match. However, after she realized his intentions, she gently rebuffed him by saying that she couldn't live

happily away from home. Although he was deeply disappointed, he wasn't 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 in may occasion." By the fall of the following year, Hamilton turned his attention to a woman named Helen Bayly, "Whom I have long known and respected and liked." Hamilton was truly smitten and luckily 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 Biela comet, Bayly's comet, after her. Then a week later, he heard that both Helen and her elderly mother were ill, and he wrote this loving letter declaring "How gladly would I, if I were permitted, minister by your sick bed and try to soothe and comfort you." Helen recovered quickly 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 very loving marriage. For example, in June of 1833, when Hamilton learned that he had to wait over the weekend to return to his bride after a meeting, he sent her a letter of love and a poem that ended with how "The vow that gave me Helen, gave me peace and balm." 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 is Hamilton." By May 10th, 1834, Helen Hamilton gave birth to their first child named William Edwin Hamilton, and a little over a year later on August 4th, 1835, a boy named Archibald. However, by the time she

had her third child in 1840, to girl named Helen Eliza, things went downhill with Helen's health. Helen was diagnosed with a nervous illness, whatever that means. Hamilton wrote a friend, "My anxiety about Lady Hamilton's health has made me very unfit for writing for many months past." For the next year and a half, Helen was sent to

recuperate with her sisters and the young family struggled without her. Helen eventually returned home in January, 1842, to everyone's delight, although she did have one other six month spell in 1856.

Despite her health issues, Helen Hamilton and William Rowan Hamilton remained devoted to each other for the remainder of their lives. As Hamilton's biographer, Robert Graves, who knew both of 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." Anyway, it was on October 16th, 1843, about two years after Helen returned from that long sickness, when Helen and William Hamilton went on a walk.

Now the reason I mentioned this walk, is on this walk Hamilton had an epiphany about quaternions and so now I can finally get to what are quaternions and how do they work? Part four: Quaternions! After his paper on 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 two imaginary axes, i and j.

As he wrote his friend, Robert Graves, "Since the square root of -1 is in a certain well-known sense, a line perpendicular to the line 1, it seems natural there should be some other imaginary to express a line perpendicular to both the former. Calling the old route, as the Germans often do, i, and the new one j, I inquired

what laws ought to be assumed for multiplying a + ib + jc and x + iy + jz." However, Hamilton was a little stumped on how to multiply them. Writing, "But what are we to do with i times j?" He tried making i times j equal to 1 or equal to -1, but neither of these choices worked so that the sum of the

squares of the coefficients in the product equaled the sum of the squares of the fact. In fact, in order to make the multiplication work out, he had to make ij equal to -ji.

Now of course, he could have just defined i X j = j X i = 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 ij = k and ji = -k. With the idea that he'd somehow prove whether k had to be 0 or not.

I would like to take a moment to acknowledge what an extraordinary and revolutionary idea this was. Hamilton was working at least seven years before the invention of matrices, where a X b doesn't equal b X a. In fact, before Hamilton, there was no formal mathematics where the order of operations mattered.

Nowhere where i X j didn't equal j X i. But Hamilton just thought it could happen and tried to prove it would exist, but the problem was he still couldn't make it work out and by September 1843, it became kind of an obsession. According to Hamilton, his sons, who were just eight and nine 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 his head, "No, I can only add and subtract them." But as he was walking with

his wife along a canal, it just came to him. With his definition of k, he could show that just like i squared and j squared, k squared also equals -1. Let me show you. Since k = ij and -ji, he could define k squared

as ij X (-ji). which is i X -j squared X i. But since j squared = -1, then -j squared = positive 1, so k squared = i squared = -1. Hamilton then realized that if i squared and k squared and k squared all equaled -1, maybe k was a third imaginary axis.

His system would therefore have four parts, a real part and three imaginary directions, i, j and k. The fact that it is four parts is why Hamilton instantly called it a quaternion, quarter for four parts.

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. With this setup, Hamilton then needed a way to multiply k X i and k X j, but he realized that with his definition of k and the knowledge that

i squared and j squared and k squared equals -1, this was easily done. As he put it, "I saw that we had probably ik = -k, because ik equals i X ij and i squared = -1.

In reverse, if you multiply k X i and use j = -ji, you get ki = -ji X i, which equals j X -i squared, which equals j. So that in symmetry with the definition of k, which equals both ij and

-ji, j = both ki and -ik. Also, jk equals j X -ji = -j squared times i, which equals i, and kj = ij X j, which equals ij squared = -i. So i equals both jk and -k X j.

In total, Hamilton's rules are i squared = j squared = k squared = -1, and equations i = jk =-kj, j = ki = =ik, and k = ij = -ji. Hamilton then, immediately simplified this into a single statement, i squared = j squared = k squared = ijk = -1. Note that all six equations relating i, j and k can be found by multiplying

ijk= -1 by different letters and then using the fact that any of the letters squared equals -1. You should try it out, it's pretty cool. Hamilton was overwhelmed with excitement about the discovery, and as he told his son Archibald, "I could not resist the impulse to cut with a knife on the stone of the bridge as we passed it.

The fundamental formula with the symbols ijk, namely, i squared = j squared = k squared = ijk = -1. 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 four part quaternions, just as he had rules to multiply two-part couplets. By July 1846, he added some new vocabulary to the quaternion. He called the real term the scaler and he called the terms

with i, j, or j, the vector. Note that initially, Hamilton wasn't thinking of a vector as we do now, something with values in the x, y or Z directions, but merely as a way of distinguishing the i, j, k terms with the scaler terms. However, he did recognize

that you could take something with values in the X, Y, and Z direction, i.e. the modern definition of a vector, and describe it as a quaternion without a scaler component. W = Xi + Yi + Zk. And you could multiply it by

another vector lowercase w, as another quaternion with no scaler component and you would get a result with nine terms, which you can think of having two results. A scalar, which is the negative of the X components multiplied plus a Y components, multiplied plus a Z components multiplied, and a more complicated vector result, which is (Yz -Zy)i + (Zx - Xz)j +(Xy-Yx)k But that's not all, in October 1847, Hamilton decided that since physicists often used the operator it made sense to him to

make a quaternion operator that was defined as Note that he kept on changing the orientation of the triangle, but we stuck with the triangle facing up. The reason Hamilton did that is because in that case the triangle squared would give a result with only a scaler component and no vector component, Note that I'm gonna call this the Del Operator because that is one of the many names it was eventually given.

Also notice, that because this Dell squared operator is a scaler, one can use it without quaternions at all, which James clerk Maxwell did without the negative sign in 1864. Anyway, Hamilton noted in 1847 that, "Perhaps not less remarkable nor having less extensive consequences," is that if you use this function times any other vector (t, u, v), you would get a quaternion where the scaler part is

the negative derivative of the x component, dx, plus the y component, dy, plus the derivative of the Z component, dz, and the vector component is: Hamilton explained his logic a little bit more in this book, "Lectures on Quaternions" written in 1853, were headed that this

operator must yet become extensively useful in the mathematical studies of nature, especially with regard to the modern researchers in analytic physics, respecting attraction, heat, electricity and magnetism. Hamilton's statements about

electricity and magnetism were particularly prescient because as soon as James Clerk Maxwell read his friend Peter Tait's book on quaternions written in 1867, Maxwell started to find lots of uses for it in his magnetoelectric equations. And soon, Maxwell started adding new names to all these functions. For example, he named the scaler of Del v the Convergence of the function, as he showed how the function changed as it converged on a point.

In addition, he called the vector Del v the Twirl or perhaps the curl of the function, as it represented how a function changed when curling or twirling around a point. These functions were so helpful that Maxwell, who created his magnetoelectric equation without vectors at all, published a book restating his theories of electricity, i.e.

Maxwell's equations, using quaternions and introduced the world to the concepts of convergence and curl. It was that book that inspired a telegraph operator named Oliver Heaviside to quit his job and devote himself to understanding Maxwell's equations.

By December 2nd, 1882, Heaviside decided that although quaternions are a very remarkable system of mathematics, he found the operations too difficult and complicated to use regularly. However, Heaviside decided it would be useful to use Maxwell's ideas of convergence and curl just without the quaternions, i.e.

to define the convergence and to define the curl separately for each direction. Then in April, 1883, Heaviside decided to remove the negative sign from the convergence and call it the divergence, writing, "We may as well use the term divergence for the same quantity with a positive sign prefixed, so if the amount of

divergence be positive," it indicated positive electricity. Therefore, the divergence of W would be dX/dx + dY/dy + 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 scaler products done away with." It's only a year after Heaviside defined the divergence that American mathematician and scientist, named Josiah Willard Gibbs, put together a little pamphlet for his students titled "Elements of Vector Analysis." That took Heaviside's method 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 vector." In other words, he made a way of using Hamilton's quaternions without having to think in fourth dimension imaginary space. In Gibbs's 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 -1. Gibbs then created two ways of multiplying vectors, either with what he

called the direct product, which we now call the dot product, or what he called the skew product, which we now call the cross product, which were inspired by the scaler and vector results of quaternion multiplication. Recall that if you have a quaternion, W = Xi + Yj + ZK and multiply it to another quaternion, lowercase w, the scaler result is the negative multiplication of the X components + the Y components + the Z components.

Now compare that to Gibbs direct, or dot product, which only differ by the removal of the negative sign. In addition, the vector result of multiplying two quaternion vectors is this, which is exactly the same as Gibb's result for the skew or cross product.

Gibbs also realized that he could relate Heaviside's divergence and curl functions to Hamilton's Del v operator and the dot product or the cross product. That's why he wrote, "Dell.w is called the divergence of w and Del cross w is the curl." Where the divergence is dX/dx + dY/dy + dZ/dz and the curl is which continues to be the definition of divergence and curl.

You can clearly see from this how Gibbs's work was inspired by Hamilton's quaternions, but you can also probably see how Gibbs' result was slightly easier to deal with, especially when it came to surface integrals. However, Gibbs did not publish his book, although he did send a

copy to Heaviside in 1888. So it's mostly ignored until Gibbs's student, Edwin Bidwell Wilson, published a book on vector analysis in 1901, which was a textbook founded upon the lectures of J. Willard Gibbs.

Anyway, it was only a few years before Gibbs's death with the publication of his student's book. Scientists and mathematicians slowly started to incorporate Gibbs's

notation for vector physics 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 they were amazing, and if they wouldn't be

acknowledged in his lifetime, he was sure they would be acknowledged after his death. As he wrote in February 1854, "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." Instead, Hamilton's influence on the progression of vector calculus is almost completely forgotten, and he's mostly just remembered as a drunk who hated his wife, which brings me to... Part five, Hamilton's legacy.

Hamilton felt that his quaternions were his greatest accomplishment. And after its discovery, he spent the majority of his time working on them. Tragically, by the 1860s, his health started to deteriorate from gout and possibly overwork.

By September 2nd, 1865, after a bout of bronchitis, he died peacefully just after his 60th birthday. After Hamilton died, the Reverend, Robert Graves, who was the brother of

Hamilton's collaborator and good friend, John Graves, took 24 years to put together three books about Hamilton's life, with over 2000 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 some scientific debate. As he put it in a secret letter, "The excitement of the

conversation, the speeches, and the wine turned out to be more than I could bear, and I was seized with giddiness and rush of blood to the head, which totally incapacitated me from keeping my ideas under control." Robert Graves wrote that after this event, his other brother, the Reverend, Charles Graves, visited Hamilton and suggested that due the effect on his reputation, Hamilton should adopt a regimen of entire abstinence from alcoholic stimulants,

which for a while he did. 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. Many years after Hamilton's death, Robert Graves had to figure out how his hero could have rejected the council of his brother, and he decided that it was basically all because of Helen Hamilton

and her failing health. 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. See, Graves felt that

Hamilton needed governing to keep him from what Graves felt were his worst 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 Hamilton came back home 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 fire or hot chocolate late at night and forced him to sip porter to keep warm. Porter, that was according to Graves, "Fraught with inevitable danger." Note that almost every high ranking aristocrat drank tremendous amounts at dinner parties. In fact, Hamilton's successor

once quipped to a friend that he was gonna pretend to be a teetotaler just to get away from all that liquor. However, to Graves, somehow Helen Hamilton's unfortunate health issues somehow forced Hamilton to take an occasional nightcap, 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." You can see how people got the impression from these overwrought words of Graves that Hamilton was a drunk, and it was due to the failings 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, Hamilton

never loved his wife, Helen, and was instead, "Properly hooked by an ailing female, who either threw 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.

Anyway, Bell's 20 page diatribe was highly influential as it was easy to read. However, some people started to wonder how mismanaged servants could lead one to drunkenness. That's when several 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.

One of my favorite versions of this is a truly delightful song about the life of William Rowan Hamilton to the tune of Alexander Hamilton, and I'm gonna show you a few clips and you'll see what I mean. Wasn't that great? You shouldn't blame them because it was only a year after they made that video, in 2017, when the narrative started to change. That was when a Dutch

researcher, Anne van Weerden, published a book defending the reputation of both William and Helen Hamilton titled, "A Victorian Marriage." 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, and I hope this video does its part, however small, for helping her in her quest to correct this historic wrong. There is one aspect of Hamilton's life that is even more maligned

than his personal one, and that is the view of the quaternions. Because although quaternions are highly respected among mathematicians and computer scientists and programmers, it tends to be completely ignored or dismissed by most physicists.

As the author, Eric Bell, said in 1937, remember the guy who was like, Helen Hamilton is a weakling who forced Hamilton to drink? "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." The strange thing is Eric Bell should have known that Hamilton was right. Quaternions are the

key for the mathematics of the physical universe. Why are so many people taught the quaternions were replaced by vector calculus instead of inspiring vector calculus? Well, that has to do with a pretty vicious debate between Oliver Heaviside and Hamilton's protege, Peter Tait, with input from Maxwell,

Gibbs, 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

I've changed my tagline from the Lightning Tamers to the Evolution of Wireless, and that is because I finished doing videos on the history of electricity and how it got developed and put in the house, and I put it into my first book, "The Lightning Tamers," which is available wherever

fine books are sold. I'm now working on my next series of videos, which will be incorporated in the book, "The Evolution of Wireless," from Sir Isaac Newton to Hedy Lamarr, and I'm also working on my second book, which is going to be called "The Radium Revolution." So make sure to hit that subscribe and the bell thingy and all that 'cause I have

lots of great stories to tell. Also, I put the script for this video with not only citations but clickable citations on my website, www.kathylovesphysics.com, so you should check that out. A big thank you to my patrons who are always so helpful, if you wanna join their ranks, there is a link down below.

And a special thanks to Anne van Weerden for her incredible and dedicated research, as well for taking so much time to help a complete stranger get this right. I put a link to her book and website down below. Stay safe and curious my friends.

Wherever fine books are sold. I thought I did a very good job with that. Similar... similar... similar (stuttering) I can't say that- similarly.

Also, (chuckling) I just can't say similarly. Too hard for me.