Quaternion



The  number system is an extension the s of mathematics. It was first discovered by William Rowan Hamilton in 1843 and subsequently defined by him as the quotient of two directed lines in a three-dimensional space, or equivalently, as the quotient of two vectors. It is studied in pure mathematics and applied to mechanics in three-dimensional space.

Quaternions are generally represented in the form
 * $$a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k$$

where $$a, b, c, d$$ are s; and $$\ \mathbf i, \ \mathbf j, \ \mathbf k$$ are the basic quaternions. Multiplication of quaternions is.

Quaternions have current practical applications in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in, , and crystallographic texture analysis. Depending upon the application, they can be used with other methods of rotation, such as with the or, or used as an alternative to them.

William Rowan Hamilton's initial 1843 flash of discovery, as depicted on a commemorative plaque on the on was
 * $$i^2 = j^2 = k^2= ijk = -1$$.

Quotes
on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication $$i^2 = j^2 = k^2 = ijk = -1$$ & cut it on a stone of this bridge.
 * Here as he walked by,
 * Quaternion plaque on, commemorating Hamilton's discovery while on a walk with his wife along the . Mathematicians from all over the world have been known to take part in the annual commemorative walk from  to the site.


 * Quantum theory may be formulated using s over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. ...[P]roblems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". ... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. ...There are precisely four 'normed division algebras': the real numbers $$\mathbb{R}$$, the complex numbers $$\mathbb{C}$$, the quaternions $$\mathbb{H}$$ and the octonions $$\mathbb{O}$$. Roughly speaking, these are the number systems extending the reals that have an ‘absolute value’ obeying the equation |xy| = |x| |y|. Since the octonions are nonassociative [their use] proves difficult... except in a few special cases. ...[I]nstead of being distinct alternatives, real, complex and quaternionic quantum mechanics are three aspects of a single unified structure.
 * John C. Baez, "Division algebras and quantum theory" (2011) arXiv:1101.5690.


 * It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes.
 * Julian Lowell Coolidge, A History of Geometrical Methods (1940). Reference is to Hermann Grassmann, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).


 * [Q]uaternions form the appropriate algebraic basis for a description of nature whenever we have to deal either with pseudoreal group representations or with co-representations of Wigner's Type II. The context in which quaternions arose historically, in a study of the three-dimensional rotation group, can now be seen to be an extremely special case of this general principle. Every group which admits pseudoreal representations equally admits a natural description in terms of real quaternions.
 * Freeman Dyson, "The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics." (Nov-Dec 1962) Journal of Mathematical Physics, Vol. 3, No. 6, pp. 1199-1215.


 * The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of by Monge in 1795. The quaternions of Sir William Rowan Hamilton, the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acuteness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict.
 * W. M. Ferriss, "Geometry," The American Cyclopaedia: A Popular Dictionary for General Knowledge (1883) ed., George Ripley, Charles Anderson Dana, Vol. 7, p. 700. Additional Note: Hermann Grassmann, Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844).


 * Frobenius' Theorem. Over the real number field there exist precisely three associative s, namely the real numbers, the complex numbers, and the real quaternions.
 * , J. reine u. agnew, Math 84, 59 (1878), Leonard Eugene Dickson, Linear Algebras (1914), as quoted in Freeman Dyson, "The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics." (Nov-Dec 1962) Journal of Mathematical Physics, Vol. 3, No. 6, pp. 1199-1215.


 * More than a third part of a century ago, in the library of an ancient town, a youth might have been seen tasting the sweets of knowledge to see how he liked them. He was of somewhat unprepossessing appearance, carrying on his brow the heavy scowl that the "mostly-fools" consider to mark a scoundrel. In his father's house were not many books, so it was like a journey into strange lands to go book-tasting. Some books were poison; theology and metaphysics in particular they were shut up with a bang. But scientific works were better; there was some sense in seeking the laws of God by observation and experiment, and by reasoning founded thereon. Some very big books bearing stupendous names, such as Newton, Laplace, and so on, attracted his attention. On examination, he concluded that he could understand them if he tried, though the limited capacity of his head made their study undesirable. But what was Quaternions? An extraordinary name! Three books; two very big volumes called Elements, and a smaller fat one called Lectures. What could quaternions be? He took those books home and tried to find out. He succeeded after some trouble, but found some of the properties of vectors professedly proved were wholly incomprehensible. How could the square of a vector be negative? And Hamilton was so positive about it. After the deepest research, the youth gave it up, and returned the books. He then died, and was never seen again. He had begun the study of Quaternions too soon.
 * Oliver Heaviside, Electromagnetic Theory (1912), Volume III; Appendix K: Vector Analysis, p. 135; "The Electrician" Pub. Co., London.


 * My own introduction to quaternionics took place in quite a different manner. Maxwell exhibited his main results in quaternionic form in his treatise. I went to Prof Tait's treatise to get information, and to learn how to work them. I had the same difficulties as the deceased youth, but by skipping them, was able to see that quaternionics could be employed consistently in vectorial work. But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vectorial aspects antiphysical and unnatural, and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalar and vectors, using a very simple vectorial algebra in my papers from 1883 onward. The paper at the beginning of vol. 2 of my Electrical Papers may be taken as a developed specimen; the earlier work is principally concerned with the vector differentiator ∇ and its applications, and physical interpretations of the various operations. Up to 1888 I imagined that I was the only one doing vectorial work on positive physical principles; but then I received a copy of Prof. Gibbs's Vector Analysis (unpublished, 1881-4).
 * Oliver Heaviside, Electromagnetic Theory (1912), Volume III; Appendix K: Vector Analysis, p. 136; "The Electrician" Pub. Co., London.


 * Mr. McAulay asks: "What is the first duty of the physical vector analyst quâ physical vector analyst?" The answer is... to present the subject in such a form as to be most easily acquired, and most useful when acquired. ...What then is the cause of the fact ...all of us deplore? ...We need only a glance at the volumes in which Hamilton set forth his method. No wonder that physicists and others failed to perceive the possibilities of simplicity, perspicuity, and brevity... in a system presented... in ponderous volumes of 800 pages. ...[I]f we turn to his earlier papers on Quaternions in the Philosophical Magazine... we find... "On Quaternions; or on a New System of Imaginaries in Algebra," and in them we find a great deal about imaginaries and very little of a vector analysis. To show how slowly the system of vector analysis developed itself in the quaternionic nidus, we need only say that the symbols S, V, and &nabla; do not appear until two or three years after the discovery of quaternions. In short it seems to have been only a secondary object with Hamilton to express the geometrical relations of vectors... it was never allowed to give shape to his work. ...[I]s it not discouraging to be told that in order to use the quaternionic method one must give up the progress which he has already made in the pursuit of his favourite science and go back to the beginning and start anew on a parallel course? ...Whatever is special, accidental, and individual, will die, as it should; but that which is universal and essential should remain as an organic part of the whole intellectual acquisition. If that which is essential dies with the accidental, it must be because the accidental has been given the prominence which belongs to the essential. ...In Italy they say all roads lead to Rome. In mechanics,, astronomy, physics, all study leads to the consideration of certain relations and operations. These are the capital notions; these should have the leading parts in any analysis suited to the subject.
 * Josiah Willard Gibbs, Nature (Mar 16, 1893) Vol. XLVII, pp. 463-464. Also see Gibbs, The Scientific Papers: Dynamics vector analysis and multiple algebra electromagnetic theory of light, etc. (1906) Vol. 2, pp. 169-170.


 * If I wished to attract the student of any of these sciences to an algebra for vectors, I should tell him that the fundamental notions of this algebra were exactly those with which he was daily conversant. ...I should call his attention to the fact that Lagrange and Gauss used the notation (αβγ) to denote precisely the same as Hamilton by his S(αβγ) except that Lagrange limited the expression to s, and Gauss to vectors of which the length is the secant of the latitude, and I should show him that we have only to give up these limitations, and the expression (in connection with the notion of geometrical addition) is endowed with an immense wealth of transformations. I should call his attention to the fact that the notation $$[r_1r_2]$$, universal in the theory of orbits, is identical with Hamilton's V$$(\rho_1\rho_2)$$ except that Hamilton takes the area as a vector... I confess that one of my objects was to show that a system of vector analysis does not require any support from the notion of the quaternion, or... of the imaginary in algebra.
 * Josiah Willard Gibbs, Nature (Mar 16, 1893) Vol. XLVII, pp. 463-464. Also see Gibbs, The Scientific Papers: Dynamics vector analysis and multiple algebra electromagnetic theory of light, etc. (1906) Vol. 2, pp. 170-172.


 * Prof. Tait has spoken of the calculus of quaternions as throwing off in the course of years its early Cartesian trammels. I wonder that he does not see how well the progress in which he has led may be described as throwing off the yoke of the quaternion. A characteristic example is seen in the use of the symbol &nabla;. Hamilton applies this to a vector to form a quaternion, Tait to form a linear vector function. ...Now I appreciate and admire the generous loyalty toward one whom he regards as his master which has always led Prof. Tait to minimise the originality of his own work in regard to quaternions and write as if everything was contained in the ideas which flashed into the mind of Hamilton at the classic . But... we owe duties to our scholars as well as to our teachers, and the world is too large, and the current of modern thought is too broad, to be confined by the ipse dixit [he says] even of a Hamilton
 * Josiah Willard Gibbs, Nature (Mar 16, 1893) Vol. XLVII, pp. 463-464. Also see Gibbs, The Scientific Papers: Dynamics vector analysis and multiple algebra electromagnetic theory of light, etc. (1906) Vol. 2, pp. 172.


 * I certainly admit that vectors may be used in connection with and to represent rotations. I have no objection to calling them in such cases versorial. In that sense Lagrange and Poinsot... used versorial vectors. But what has this to do with quaternions? Certainly Lagrange and Poinsot were not quaternionists. ...Does it follow that I have used a quaternion? Not at all. A quaternionic expression may represent a number. Does everyone who uses any expression for that number use quaternions? A quaternionic expression may represent a vector. Does everyone who uses any expression for that vector use quaternions? A quaternionic expression may represent a linear vector operator. If I use expression for that linear vector operator do I therefore use quaternions? My critic is so anxious to prove that I use quaternions that he uses arguments which would prove that quaternions were in common use before Hamilton was born.
 * Josiah Willard Gibbs, Nature (Aug 17, 1893) Vol. XLVIII, pp. 364-367. Also see Gibbs, The Scientific Papers: Dynamics vector analysis and multiple algebra electromagnetic theory of light, etc. (1906) Vol. 2, p. 174.


 * I have not been exclusively occupied by my Quaternions, but confess that they have been growing in interest upon me, and that I more and more believe they will one day justify a hope which I ventured to express... that they will constitute nothing less than "a new algebraical geometry."
 * William Rowan Hamilton, Letter to Augustus De Morgan (Feb 2, 1846) as quoted in, The Life of Sir William Rowan Hamilton (1885) Vol. 3, p. 261.

But on the 16th day of the same month… I was walking… and your mother was walking with me, along the … and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof... I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse—unphilosophical as it may have been—to cut with a knife on a stone of, as we passed it, the fundamental formula with the symbols, i, j, k; namely, $$i^2 = j^2 = k^2 = ijk = -1,$$
 * I had been wishing for an occasion of corresponding a little with you on Quaternions: and such now presents itself, by your mentioning in your note... that you "have been reflecting on several points connected with them"... "particularly on the Multiplication of Vectors. ...No more important, or ...fundamental question, in the whole theory of Quaternions, can be proposed than that which thus inquires What is such Multiplication? What are its Rules, its Objects, its Results? What Analogies exist between it and other Operations, which have received the same general Name? And finally, what is (if any) its Utility? ...[R]eferring to an ante-quaternionic time, when you were a mere child, but had caught from me the conception of a Vector, as represented by a Triplet... I happen to be able to put the finger of memory upon the year and month—October, 1843—when... the desire to discover the laws of the multiplication referred to regained with me a certain strength and earnestness, which had for years been dormant, but was then on the point of being gratified, and was occasionally talked of with you. Every morning in the early part of the... month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, "Well, Papa, can you multiply triplets"? Whereto I was always obliged to reply, with a sad shake of the head: "No, I can only add and subtract them."
 * William Rowan Hamilton, Letter to Archibald H. Hamilton, his son (Aug 5, 1865) as quoted in, The Life of Sir William Rowan Hamilton (1885) Vol. 2, pp. 434-435.


 * all the numbers that have been derived from the genus are four; but number is the indefinite genus, from which was constituted, according to them, the perfect number, viz. the decade. For one, two, three, four, become ten, if its proper denomination be preserved essentially for each of the numbers. Pythagoras affirmed this to be a sacred quaternion, source of everlasting nature, having, as it were, roots in itself; and that from this number all the numbers receive their originating principle.
 * Hippolytus of Rome, Ante-Nicene Christian Library, Volume 6: Hippolytus, Bishop Of Rome, Vol. 1, p. 32.


 * The familiar proposition that all A is B, and all B is C, and therefore all A is C, is contracted in its domain by the substitution of significant words for the symbolic letters. The A, B, and C, are subject to no limitation for the purposes and validity of the proposition; they may represent not merely the actual, but also the ideal, the impossible as well as the possible. In Algebra, likewise, the letters are symbols which, passed through a machinery of argument in accordance with given laws, are developed into symbolic results under the name of formulas. When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power.
 * Benjamin Peirce, On the Uses and Transformations of Linear Algebra (1875) An address to the American Academy of Arts and Sciences, May 11, 1875.


 * [O]f possible quadruple algebras the one... by far the most beautiful and remarkable was practically identical with quaternions, and... it [is] most interesting that a calculus which so strongly appealed to the human mind by its intrinsic beauty and symmetry should prove to be especially adapted to the study of natural phenomena. The mind of man and that of Nature’s God must work in the same channels.
 * Benjamin Peirce, as quoted by W. E. Byerly, III. Reminiscences (1925) The American Mathematical Monthly, Mathematical Association of America, Vol. 32, p. 6.


 * [T]he common solution, using three Euler's angles interpolated independently, is not ideal. The more recent (1843) notation of quaternions is proposed instead, along with interpolation on the quaternion unit sphere. Although quaternions are less familiar, conversion to quaternions and generation of in-between frames can be completely automatic, no matter how key frames were originally specified, so users don't need to know—or care—about inner details. The same cannot be said for Euler's angles, which are more difficult to use.
 * Ken Shoemake, "Animating Rotation with Quaternion Curves" (1985) Computer Graphics, Volume 19, Issue 3, Jul. 1985 pp. 245–254.


 * While translations are well animated by using vectors, rotation animation can be improved by using the progenitor of vectors, quaternions. ...By an odd quirk of mathematics, only systems of two, four, or eight components will multiply as Hamilton desired; triples had been his stumbling block.
 * Ken Shoemake, "Animating Rotation with Quaternion Curves" (1985) Computer Graphics, Volume 19, Issue 3, Jul. 1985 pp. 245–254.


 * Closely akin to his third and fourth propositions is Riemann's fifth proposition, that continuous quantities are coördinate with discrete quantities, both being in their nature multiples or aggregates, and therefore species of the same genus. This pernicious fallacy is one of the traditional errors current among mathematicians, and has been prolific of innumerable delusions. It is this error which has stood in the way of the formation of a rational, intelligible, and consistent theory of irrational and imaginary quantities, so called, and has shrouded the true principles of the doctrine of "complex numbers" and of the calculus of quaternions in an impenetrable haze.
 * , The Concepts and Theories of Modern Physics (1881) Ch XIV. Metageometrical Space in the Light of Modern Analysis Riemann's Essay, p. 260.


 * The next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions.
 * Peter Guthrie Tait, Note on a Quaternion Transformation , Communication read on Monday, 6th April, 1863, Proceedings of the Royal Society of Edinburgh (1866), p. 117.


 * I do think... that you would find it would lose nothing by omitting the word "vector" throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.
 * William Thomson, 1st Baron Kelvin, (Dec 1892) Letter to with respect to Hayward's Algebra of Coplanar Vectors and Trigonometry (1899); as quoted by, Life of Lord Kelvin (1910) Vol. 2, p. 1138.


 * Symmetrical equations are good in their place, but "vector" is a useless survival, or offshoot, from quaternions, and has never been of the slightest use to any creature. Hertz wisely shunted it, but unwisely he adopted temporarily Heaviside’s nihilism. He even tended to nihilism in dynamics, as I warned you soon after his death. He would have grown out of all this, I believe, if he had lived. He certainly was the opposite pole of nature to a nihilist in his experimental work, and in his Doctorate Thesis on the impact of elastic bodies.
 * William Thomson, 1st Baron Kelvin (April 29, 1896) footnote in Letter to George Francis FitzGerald, as quoted by, Life of Lord Kelvin (1910) Vol. 2, pp. 1070.

I think you are right in your use of the word "Volapuk," but I don't think you should confine it to the vector part of quaternions. The whole affair has in respect to mathematics a value not inferior to that of "Volapuk" in respect to language.
 * I see no possible objection to your now publishing the deferred "scrap" if you yourself approve of what is said in it in favour of quaternions.
 * William Thomson, 1st Baron Kelvin, (July 25, 1896) Letter to Peter Guthrie Tait, as quoted by, Life of Lord Kelvin (1910) Vol. 2, pp. 1137-1138.


 * I first became personally acquainted with Tait a short time before he was elected Professor in Edinburgh… It must have been either before his election or very soon after it that we entered on the project of a joint treatise on Natural Philosophy. He was then strongly impressed with the fundamental importance of Joule’s work… We incessantly talked over the mode of dealing with energy which we adopted in the book, and we went most cordially together in the whole affair. … We have had a thirty-eight years’ war over quaternions. He had been captivated by the originality and extraordinary beauty of Hamilton’s genius in this respect; and had accepted, I believe, definitely from Hamilton to take charge of quaternions after his death, which he has most loyally executed. Times without number I offered to let quaternions into Thomson and Tait if he could only show that in any case our work would be helped by their use. You will see that from beginning to end they were never introduced.
 * William Thomson, 1st Baron Kelvin, (July 13, 1901) Letter to G. Chrystal, as quoted by, Life of Lord Kelvin (1910) Vol. 2, pp. 452-453.


 * Yet, though few, if any—Clerk-Maxwell perhaps only excepted—ever possessed the same almost magical quality of physical insight, none could be more strict than Lord Kelvin in requiring demonstration freed from untenable assumptions or undemonstrable hypotheses. Daring as he was, at least in his earlier days, in the application of analytical methods to the phenomena of nature, he was in several ways very conservative. For example, he never would countenance the use in physics of the method of quaternions. At the British Association Meeting at Cambridge in 1845, he had met Hamilton, who there read his first paper on Quaternions. One might have thought that the young enthusiast would have readily welcomed a new and ingenious method of symbolic analysis: but it was not so. He would not use quaternion notation or quaternion methods himself, nor did he admit the into his work.
 * , Life of Lord Kelvin (1910) Vol. 2, pp. 1137.

Elementary Sketch of the Nature of that Conception of Mathematical Quaternions (1889)

 * Which is Developed More in Detail by Sir W. R. Hamilton, in his Recently Published Volume of Lectures on that Subject. as quoted in Appendix to, The Life of Sir William Rowan Hamilton (1885) Vol. 3, pp. 635-637.


 * '(1). The word "Quaternion" requires no explanation, since... it occurs in the Scriptures and in Milton. Peter was delivered to "four quaternions of soldiers" to keep him; Adam, in his morning hymn, invokes air and the elements, "which in quaternion run." The word (like, the Latin "quaternio," from which it is derived) means simply a set of four, whether those "four" be persons or things.


 * '(2). But the question arises, what special connexion has the number Four with mathematics generally, or with that branch of mathematical science in particular, to which the "Lectures on Quaternions" relate?

Might in the Chain of Symbol girdled be."
 * '(3). One general form of answer... is... that in the mathematical quaternion is involved a peculiar synthesis, or combination, of the conceptions of space and time; and that while TIME is usually pictured or represented by metaphysicians under the figure of a line—a single stream with its ONE current—an unique axis of progression, SPACE is, on the contrary, imagined or conceived in connexion with THREE distinct axes, three lines at right angles to each other... height, length, and breadth. In time, we have only the forward and the backward, looking before and after. In space, there is not merely the contrast between the directions of upward and downward, but also between those of southward and northward, and again between westward and eastward. Time is said to have only one dimension, and space to have three dimesions. The former is an unidimensional, the latter a tridimensional progression. The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space," or "space plus time": and in this sense it has, or at least it involves a reference to, four dimensions. In an unpublished sonnet to Sir John Herschel, entitled "The "(...Greek ...equivalent to the Latin Quaternio), the author of the Lectures introduced the two following lines... an expression of the view... in the foregoing remarks..: "And how the One of Time, of Space the Three,


 * '(4). Those who are entirely unacquainted with mathematical science may yet derive, from what has been above remarked, a sufficient preliminary insight into the nature of the speculations and inquiries to which the "Lectures on Quaternions" relate. A philosophical, if not a technically scientific, knowledge of the author's general aim, and of the idea which has guided him, may in this way be easily attained. But a very moderate acquaintance with the conceptions of geometry will suffice to render intelligible, from another point of view, the importance which the author attaches to the number Four in mathematics.


 * (5). As early as the first book of Euclid's Elements, an attentive student is (or may be) led to consider the relative length, and also the relative direction, of one straight line as compared with another. Thus when Euclid shows, in his very first proposition, how to construct on a given base AB an equilateral triangle ABC, he virtually teaches how, when one line AB is proposed or given, to draw a new line BC (or AC), which shall in length be equal to the given one, and in direction shall make with it an angle of sixty degrees, namely, the angle ABC (or BAC), which is the third part of 180 degrees, or of two right angles.


 * '(6). In this elementary example, if the length of the given base AB be taken as the standard of length, and be on that account called unity, or one, then the length of the side BC (or AC) of the triangle must also be denoted by the same number, ONE ; and these TWO NUMBERS, one, and sixty, serve in this view to define, or to describe, the length and direction of the new or constructed line BC; at least if the latter number (sixty) be combined with the consideration of a certain hand, or direction of rotation, towards which the old line BA may be conceived to turn, in the plane of the triangle (or of the paper)...


 * '(7). The foregoing view, although not precisely the same with that adopted by Euclid himself, in his exposition of the elements of geometry, is at least consistent therewith; and has been made the basis of an important and modern method of calculation, respecting directed lines in one plane, which seems to have been first introduced about the commencement of the present century, by Argand in France, and for which Professor De Morgan... has lately proposed the name of Double Algebra: because it recognises and employs two numerical elements (such as the numbers 1 and 60 in the foregoing example), as required for the joint determination of the length and direction of a straight line. And it is now to be shown what is the nature of the passage that has been made, by the author of the Lectures on Quaternions, from such a double system of algebraic geometry, to what may be called, by analogy and contrast, a quadruple system of calculations respecting directed lines, or a system of Q UADRUPLE A LGEBRA.
 * Ref: Augustus De Morgan, Trigonometry and Double Algebra (1849)


 * '(8). This passage from the one system to the other may be said to consist mainly in the consideration of the variable plane of an angle. If, after tracing the equilateral triangle ABC on a card, which at first rests on a horizontal table, we then lift up that card, with the figure traced thereon, and lay it on a sloping desk, the triangle in its new position takes also a new aspect; it faces a different region of space, and may be conceived to look at, or be looked at by, a new point of the heavens, which is not now the vertical point (or ), as before. This new aspect of the figure, or of the plane (or desk) on which it is now situated, is the new circumstance introduced, in the transition from Double to Quadruple Algebra. And in fact it is easy to see that this new circumstance, of the varied position of the figure, namely, of the triangle, or simply (if we choose) of the ANGLE ABC, requires the consideration of two new numerical elements. For we have now two new questions to answer, or two new things to determine: namely, 1st, the slope of the desk (or inclination of the plane), suppose forty-five degrees, conducting to a first new number, 45 ; and 2nd, the direction of the edge (or, technically speaking, the line of the nodes), where that slope meets the table, and which may deviate from the line of north and south by any other number of degrees, suppose seventy, giving thus a second new number, in this case 70.'