Hamilton, the Quaternions, and Mathematical Mythologies

The arduous discovery of the properties of the quaternions by William Rowan Hamilton has always stuck in my mind as among the most romantic of modern math’s encounters. Until the fall of 1843, Hamilton was set to work extending the complex numbers, by then a regular feature of the theory of equations. As he would later write in a letter to his son,

Every morning in the early part of October 1843, on my coming down to breakfast, your brother William Edwin and yourself used to ask me: “Well, Papa, can you multiply triples?” Whereto I was always obliged to reply, with a sad shake of the head, “No, I can only add and subtract them.”

Lucky for us, the mathematician was no more than a few weeks from perspiration paying a handsome dividend of inspiration. The image topping this post is an inscription, set by the city of Dublin, on Brougham Bridge, which reads,

Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i2 = j2 = k2 = ijk = −1 & cut it on a stone of this bridge.

William Rowan Hamilton - Wikipedia
A portrait of Sir WR Hamilton, ca. 1860.

I used to read fondly about the Hamilton Walk some lucky mathematicians and quaternion admirers went on annually, along the foggy span of the Royal Canal, retracing his eureka moment.

Imagine my surprise, then, when in the course of reading Tristan Needham’s peerless Visual Complex Analysis, I read a note on the history of the quaternions which alluded to prior discovery.

As is well known, the quaternion rule was discovered in algebraic form by Sir William Rowan Hamilton in 1843. It is less well known that three years earlier Olinde Rodrigues had published an elegant geometric investigation of the composition of rotations in space that contained essentially the same result…

Needham, Visual Complex Analysis, p. 44
Olinde Rodrigues - Wikipedia
A portrait of Olinde Rodrigues, date unknown.

Rodrigues was an interesting figure in his own right. His mathematical discoveries were noteworthy, but the greater part of his writings were political. He was a noted follower of the Comte de Saint-Simon, whose philosophy, Saint-Simonianism, counts among its descendants the utilitarianism of Mill, the anarchism of Proudhon, and the positivism of Auguste Comte. Despite all that, Rodrigues was said to be Saint-Simon’s favorite.

His works on math, though, are what we’re dealing with here. They shift the perspective on quaternionic discovery. Far from being the classic eureka moment of 1840s Irish mathematics, the near-simultaneous discovery sets up a new battle in the mold of the great Newton-Leibniz Calculusschlacht.

Of course, that would be the case, had both Rodrigues and Hamilton not been preempted by – who else? – Gauss:

Hamilton and Rodrigues are just two examples of hapless mathematicians who would have been dismayed to examine the unpublished notebooks of the great Karl Friedrich Gauss. There, just like another log entry in the chronicle of his private mathematical voyages, Gauss recorded his discovery of the quaternion rule in 1819.

Needham, Visual Complex Analysis, p. 44

I was motivated to track this citation down and stumbled upon a kindly scan of Gauß’ wissenschaftliches Tagebuch 1796-1814, published in 1903 and edited by none other than Felix Klein.

Klein himself was among the greatest of the many great mathematicians working in Göttingen and others of the first-class German universities in the tail end of the long nineteenth century; he studied under Lipschitz and taught, among others, Gregorio Ricci-Curbastro (developer of the theory of tensor calculus), Walther von Dyck (formalizer of the modern notion of group), and Max Planck. In 1875 he married Hegel’s granddaughter, Anne, and in 1895 he hired David Hilbert as a professor at Göttingen. His was the recommendation that the basis of secondary-level mathematical education be analytic geometry, which endures today in the common Algebra -> Geometry -> Pre-Calculus sequence.

Felix Klein
Felix Klein, by Max Lieberman, 1912. In the collection of the Mathematisches Institut Göttingen.

Aside from the Klein bottle, Klein is best known for his lectures at Erlangen, the first place he held a professorship (at 23!), wherein he motivated a new synthesis for modern mathematics and established new directions for mathematical research – this is the famous Erlangen Program. Needham puts it well:

…[A] geometric property of a figure is one that is unaltered by all possible motions of the figure…in answer to the opening question of “What is geometry?”, Klein would answer that it is the study of these so-called invariants of the set of motions…Klein’s idea was that we could first select a group G at will, then define a corresponding “geometry” as the study of the invariants of that G.

Needham, Visual Complex Analysis, p. 32, 33

Amid all that work, Klein found the time to edit Gauss’ journals. May we all be so productive. In his preface, he noted one of two things “which bestow upon the journal incomparable biographical value” :

What we win for ourselves here is an unadulterated, personable insight into the scientific development of the young Gauss in the years 1796-1800.

[Diese] ist der unmittelbare, sozusagen persönliche Einblick, den wir gerade für die entscheidenden Jahre 1796-1800 in den wissenschaftlichen Werdegang des jungen GauB gewinnen.

Klein, Preface to Gauß’ wissenschaftliches Tagebuch, 1796-1814, p.2

The reader at this stage may have noticed the regrettable error in my research – the copy of Gauss’ journals I could find on the internet only went through 1814. The note in which he sets out the defining relations for the quaternionic algebra appeared in 1819. Try as I might, I ended up there, and so cannot get to the text of what Gauss did find.

Wikipedia stakes a further claim in the really nicely done “History of Quaternions” article. It relates that the quaternionic relationship is implicit in Euler’s four-squares identity, which I believe, but cannot work out for myself.

Regardless of the ultimate decision on who found quaternions first, I feel glad to have tracked down a little of the mathematical lives which worked on this problem and admired each other’s work.