It occurs to me that in 1830 or so Gauss published his half baked ideas on imaginary aithmetiks, unsure of the metaphysical ground for them, but inncreasingly irked by the french progress in this field by Argand, De Cauchy ad Galois. Fourier also was beginning to investigate heat conduction using Newton's ideas and roots of unity also a Newtonian idea extensively demonstrated by De Moivre and Cotes. Having found Wessels paper cogent and persuasive, Gauss was emboldened to place his ideas into the ring. Nothing was certain.
Hamilton's friend from college days, John T. Graves, had developed a theory of imaginary logarithms, which was published in the Philosophical Transactions of the Royal Society of London for the year 1829. This theory was not generally accepted, and George Peacock (1791-1858) had criticised it in his Report on Algebra to the British Association for the Advancement of Science in 1833
Meanwhile a friend of Hamilton was emboldened by the turmoil to demonstrate the relationship between the most practical calculating system ever devised, the logarithms and the imaginary values. He received such a round rebuke that his name was besmirched and calumnied throughout "Mathdom", the invisible kingdom that mathematicians inhabit. Hamilton set upon the task of vindication of his friend.
To give the popular reader an idea of the nature of Quaternions, and the steps by which Hamilton was, during some fifteen years, gradually conducted to their invention, it is necessary to refer to the history of a singular question in algebra and analytical geometry, the representation or interpretation of negative and imaginary (or impossible) quantities.
Descartes' analytical geometry and allied methods easily gave the representation of a negative quantity. For it was seen at once to be a useful convention, and consistent with all the fundamental laws of the subject, to interpret a negative quantity as a quantity measured in the opposite direction to that in which positives of the same kind are measured. Thus a negative amount of elevation is equivalent to depth, negative gain is loss, a negative push is a pull, and so on. And no error, but rather great gain in completeness and generality, results from the employment of this convention in algebra, trigonometry, geometry, and dynamics.
But it is not precisely from this point of view that we can readily see our way to the interpretation of impossible quantities. Such quantities arise thus: If a positive quantity be squared, the result is positive; and the same is true of a negative quantity. Hence, when we come to perform the inverse operation, i.e., extract the square root, we do not at once see what is to be done when the quantity to be operated on is negative. When it is positive, its square root may be either a negative or a positive number, as we have just seen. If positive, it is to be measured off in some definite direction, if negative, in the opposite. But how shall we proceed to lay off the square root of a negative quantity? Wallis, in the end of the sixteenth century, suggested that this might be done by going out of the line on which the result, when real, would have been laid down; and his method is equivalent to this:- Positive unity being represented by an eastward line, negative unity will of course be represented by an equal westward line, and these are the two square roots of positive unity. According to Wallis' suggestion a northward and a southward line may now be taken to represent the two square roots of negative unity, or the so-called impossibles or imaginaries of algebra. But the defect of this is that we might have assumed with equal reason any other line (perpendicular to the eastward one) as that on which the imaginary quantities are to be represented. In fact, Wallis' process is essentially limited to plane problems, and has no application to tridimensional space. But, imperfect as this step is, it led at once to another of great importance, the consideration of the length, and direction, of a line independently of one another. And we now see that as the factor negative unity simply reverses a line, while the square root of negative unity (employed as a factor) turns it through a right angle, the one operation may be looked upon as in a certain sense a duplication of the other. In other words, twice turning through a right angle, about the same axis, is equivalent to a reversal; or, negative unity, being taken to imply reversal of direction, may be considered as rotation through two right angles, and its square root (the ordinary imaginary or impossible quantity) may thus be represented as the agent which effects a certain quadrantal rotation. But, as before remarked, the axis of this rotation is indeterminate; it may have any direction whatever perpendicular to the positive unit line. If we fix on a particular direction, everything becomes definite, and we can on the same plan interpret the (imaginary) cube roots of negative unity as factors or operators which turn a line through an angle of sixty degrees positively or negatively. Similarly, any power of negative unity, positive or negative, whole or fractional, obtains an immediate representation. And the general statement of this proposition leads at once (but not by the route pursued by its discoverer) to what is called De Moivre's Theorem, one of the most valuable propositions in plane trigonometry. Warren, Argand, Grassmann, and various others, especially in the present century, vainly attempted to extend this process to space of three dimensions. The discovery was reserved for Hamilton, but was not attained even by him till after fifteen or twenty years of arduous work. And it is a curious fact that it was by speculations totally unconnected with geometry that he was so prepared as to see, almost at the instant of seizing it, the full value of his invention. The frightful complexity of the results to which Warren was led in endeavouring to express as lines the products and quotients of directed lines in one plane, seems to have induced Hamilton to seek for a representation of imaginary quantities altogether independent of geometry. The results of some at least of his investigations are given in a very curious essay, Algebra as the Science of pure Time, communicated to the Royal Irish Academy in 1833, and published, along with later developments, in the seventeenth volume of their Transactions. We quote considerable portions of the introductory remarks prefaced to this Essay, as they show, in a very distinct manner, the logical character and the comprehensive grasp of Hamilton's mind.
Hamilton's ideas were in a way supported by Gauss's tentative work, and Graves was vindicated. Grassmann however was sidelined because he was not a recognised scholar. His insight was ignored as trifling, Hamilton,surely(?), had covered this in his great analysis.
It seems strange to report, that in the practice and study of Geometry, the notion of rotation is absent, being replaced it seems somewhat inadequately by the static notion of the circle!
Well, no. He hadn't actually. Grassmann had hit upon the importance of invariance(Keine Abweichung) in notation in relation to the dynamic pushing, and shoving and lineal motion of points and Sides. And the relationship between points and sides he perceived as Strecken, streaks of motion from one point to another. Dynamic not static, thus Length and independent Direction, not just length.
Grassmann's insight was fitful, periodic attention driven. He published nothing of the Ausdehnungslehre until 1844 after years od study and validation and refinement. He was not in any hurry or under any academic pressure nor in competition with any one. His ideas epresented his personal interaction with space. He had noticed ivariance in a system of 3 points and 3 connected strecken. The relationship was clearly cyclic. rotation about a chosen one ofthe points was involved. For ease of concept he chose B for point A, B, C.
Now when he least expected it some observations his dad, J Grassmann, was making with regard to trigonometric ratios and space theory seemed to show the same invariance for the 4 cyclic points in the plane under area multiplication, But you had to apprehend the forms differently! They were not lengths to be calculated, but streaks of motion butting up against each other and Producing a form. In particular, the euclidean notion that a rectilinear form can be referred to by its 2 sides (zwei Seiten) meant the form was a product of 2(zweier). And these 2 were henceforth in Grassmann's mind transformed into Strecke, streaks of motion, as one point streaks toward the other.
How was that represented in terminology? What terminological patterns could be seen when representing Strecken not lengths solely? We shall see.But be it known, Hamilton never saw this invariance in terminology, he used terminology in the standard way to represent an unknown or variable length only not a strecken. And in his seminal work on couples, in which he identifies the distinction between direction and length, and indeed his notation for a step is in everyway equivalent in conception to a strecken, yet he did not identify in his earlier relations the invarianc of terminology for 3 points A, B, C because he dealt with Couples: A-B, and C-D!
So now when he later came to deal with Triples, of which we have little information, his work should bear direct comparison with where Grassmann starts. Thus in shor Hamilton developed couples, Grassmann developed triples!
In 1835 Hamilton seems to have extended the above theory from Couples to Triplets, and even to a general theory of Sets, each containing an assigned number of time-steps. Many of his results are extremely remarkable, as may be gathered from the only published account of them, a brief notice in the Preface to his Lectures on Quaternions. After having alluded to them, he proceeds: `There was, however, a special importance to the consideration of triplets…. This was the desire to connect, in some new and useful (or at least interesting) way, calculation with geometry, through some undiscovered extension, to space of three dimensions, of a method of construction or representation which had been employed with success by Mr. Warren (and indeed also by other authors, of whose writings I had not then heard), for operations on right lines in one plane: which method had given a species of geometrical interpretation to the usual and well-known imaginary symbol of algebra.' After many attempts, most of which launched him, like his predecessors and contemporaries, into a maze of expressions of fearful complexity, he suddenly lit upon a system of extreme simplicity and elegance. The following remarkable interesting extract from a letter gives his own account of the discovery:-
` Oct. 15, '58.
`P.S. – To-morrow will be the fifteenth birthday of the Quaternions. They started into life, or light, full grown, on the 16th of October 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge, which my boys have since called the Quaternion Bridge. That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; exactly such as I have used them ever since. I pulled out, on the spot, a pocket-book which still exists, and made an entry, on which, at that very moment, I felt that it might be worth my while to expend the labour of at least ten (or it might be fifteen) years to come. But then, it is fair to say that this was because I felt a problem to have been at that moment solved,- an intellectual want relieved,- which had haunted me for at least fifteen years before.
`Less than an hour elapsed, before I had asked and obtained leave, of the Council of the Royal Irish Academy, of which Society I was, at that time, the President,- to read at the next general Meeting, a Paper on Quaternions; which I accordingly did, on November 13, 1843.
`Some of those early communications of mine to the Academy may still have some interest for a person like you, who has since so well studied my Volume, which was not published for ten years afterwards.
`In the meantime, will you not do honour to the birthday, to-morrow, in an extra cup of – ink? for it may be obsolete now to propose XXX,- or even XYZ.'
Euclids Stoikeioon Book 2 particularly informs Grassmanns basic observations about Strecken.
Hiermit war denn der erste Schritt zu einer Analyse gethan, welche in der
Folge zu dem neuen Zweige der Mathematik führte, der hier vorliegt. Aber
keinesweges ahnte ich, auf welch' ein fruchtbares und reiches Gebiet ich hier
gelangt war; vielmehr schien mir jenes Ergebniss wenig beachtungswerth, bis
sich dasselbe mit einer verwandten Idee kombinierte . Indem ich nämlich den
Begriff des Produktes in der Geometrie verfolgte, wie er von meinem Vater
aufgefasst wurde, so ergab sich mir, dass nicht nur das Rechteck,sondern auch
das parallelogram überhaupt als Produkt zweier an einander stossender Seiten
desselben zu betrachten sei, wenn man nämlich wiederum nicht das Produkt
der Längen, sondern der beiden Strecken mit Festhaltung ihrer Richtungen
auffasste. Indem ich nun diesen begriff des Produktes mit dem vorher
aufgestellten der Summe in Kombination brachte, so ergab sich die
auffallendste Harmonie;wenn ich nämlich, statt die in dem vorher angegebenen Sinne genommene
Summe zweier Strecken mit einer dritten in derselben Ebene liegenden
Strecke in dem eben aufgestellten Sinne zu multipliciren, die Stücke einzeln
mit derselben Strecke multiplicirte, und die Produkte mit gehöriger
Beobachtung ihrer positiven oder negativen Geltung addirte, so zeigte sich,
dass in beiden Fällen jedesmal dasselbe Resultat hervorging und hervorgehen
musste. Diese Harmonie liess mich nun allerdings ahnen, dass mich hiermit
ein ganz neue Gebiet der Analyse aufschliessen würde, was zu wichtigen
Resultaten führen konnte. Doch blieb diese Idee, da mich mein Beruf in
andere Kreise der Beschäftigung hinein zog, wieder eine ganze Zeit lang
ruhen; auch machte mich das merkwürdige Resultat anfangs betroffen, das
für diese neue Art des Produktes zwar die übrigen Gesetze der gewöhnlichen
Multiplikation und namentlich ihre Beziehung zur Addition bestehen blieb,
dass man aber die Faktoren nur vertauschen konnte, wenn man zugleich die
Vorzeichen umkehrte( + in – verwandelte und umgekehrt).
It follows,that by now the first step to an analysis was done, which in Consequence to the new branches of mathematics led, which is available here. But by no means did I guess at what a rich and fertile territory here I
had reached, but rather that result seemed to me worthy of little attention until the very same I combined with a related idea. Namely, In which I followed, the Terminology of the product in geometry, as it would be apprehended by my father; thus was revealed to me that not only the rectangle, but also above all the parallelogram, is to be considered as of the same things: a "product" of two sliding against one another sides, namely if one apprehends once again not the product of the lengths, but the two Streaks with their directions firmly attached. By use of which this terminology of the product i now brought in combination with the already established Sum terminology, so the most striking harmony revealed itself; namely, if , instead of using as in the previously given Experience the resultant Sum of two Streaks, i use a third lying in the same plane Streak in order to evaluate multiplication in line with the Experience just plainly set out, if i individually multiply the pieces by the same Streak, and add together the products with associated Observation of their positive or negative value , thusly it was shown, that in both cases every time the same result had emerged and had to emerge. This harmony let me now suspect, however, that I would unlock hereby an entirely new field of analysis , which could lead to important Results . Then this idea remained rested, since my profession in other "Circuits of employment" dragged me down and in, again for quite a while ; also the curious result made me concerned initially,the result which for this new type of product truly kept setup the other laws of the ordinary Multiplication and especially, their relationship to addition,
but that one could only swap the factors if one their respective Signs reversed at the same time (+ trasfomed to – and vice versa)
I think this means : in any parallelogram ABCD any 3 points, say ABC mark out 2 colliding Strecken AB and CB. Replacing these 2 colliding Strecken by a third in the same plane by using the invariant rule for 3 Strecken, AB + BC = AC we obtain AB + -CB = AC. So using the 2 colliding sides in the defined sense reveals that a negative value is required for one of the Strecken. Let us refer to the strecken as
c + -a = b.
We replace c and -a by b. Now we use this 3rd Strecken to multiply, and i multiply each of the pieces (of the flat figure) by this b as we would multiply the sides of a parallelogram or rectangle, and add the "products" according to the sign values and the Strecken form.
b*b = b*c + b*-a = (c + -a)*c + (c + -a)*-a = c*c + -a*c + c*-a +-a*-a
Observing the sign rules (why?)
b2 = c2 + -a*c + c*-a + a2
then we get
c2 + (-a)2 = b2
2 remarkable results that i would describe as a remarkable harmony.
This important passage in Grassmann's !842 Vorrede is difficult to elucidate in translation, not the least of it being material that Hamilton in 1833-34 seemingly covered in his Submission on Couples. However the treatment is different, and it is that slight difference that makes all the difference. Hamilton as i earlier pointed out made geat headway with 2 and 4 points, but no new insight beyond Mr Warrens with 3. Grassmann starts with 3 and from there finds path to subsume all points quite extensively! The core of this achievement was that he was paused by interest and circumstance to look at this trichotomous set of relationships and lucky enough to make the observation that if a variation applied to one and the same thing, in this case the sides of rectilinear forms then those variations may be equated. That is in rigorous presentation they form a proportion in which the simplest resolution is that they are identical.
However confounding it was that this could not be, yet he held his nerve because every law of multiplication was upheld including the relation to addition, as aid out in Euclid's fundamental teaching material. The significance of this was that regardless of what these strecken might be used to represent, the combinatorial laws were upheld,something symbolic logicians if no other would have appreciated. The point is that Grassmann was sufficiently informed on symbolic reasoning to recognise that this remarkable harmony provided a sound basis for further development, but not just in terms of lengths but symbolic descriptors of dynamic variable magnitudes called Strecken/Streaks. Hamilton, hurrying on to demonstrate the usual relations passed this by, not as overlooked, but as not at this time worthy of attention, an opinion Grassmann held at step one!
Strecken therefore were transformed as soon as they were born! They became symbols of other things that held this relationship, and then they were subsumed in symbols that extended them to other forms not designable by the term Strecken. Soon new terms like Flacheraum and others were developed, and on and on… the Ausdehnungs grosse were being realised.
In 1853 Hamilton read Die Ausdehnungslehre, and immediately saw himself in rivalry with Grassmann, but not at Grassmann's level yet! Thus he strove to catch up with and surpass Grassmann from thenceforward. His later works should sow sign of this, and indeed his last work was said to be made general to n dimensions by a trifling alteration of notation. This was never done, however by Hamilton, Whereas Grassmann at all times worked in this way, and knew the difficulties not to be trifling at all!