1.1. Preface to the Second Edition.

Sir William Rowan Hamilton died on the 2nd of September, 1865, leaving his great work on Quaternions unfinished. He intended to have added some account of the operator ∇, an Index, and an Appendix containing notes on Anharmonic Coordinates, on the Barycentric Calculus, and on proofs of his geometrical theorems stated in Nichol's Cyclopaedia. At the time of his death, with the exception of a fragment of the preface, and a small portion of the table of contents, all the manuscript he had prepared was in type. As he rarely commenced writing before his thoughts were fully matured, he has left no outline of the additions contemplated.

In this edition, printed by direction of the Board of Trinity College, Dublin, the original text has been faithfully preserved, except in a few places where trifling errors have been corrected. I have added notes, distinguished in every case by square brackets, wherever I thought they were wanted. I have rendered the work more convenient by increasing the number of cross-references, by including in the page-headings the numbers of the articles (for the original references are generally given to articles and not to pages), by dividing the work into two volumes, and by the addition of an index. The table of contents has been amplified by a brief analysis of each article, designed as far as possible to assist the reader in following and in recapitulating the arguments in the text. Hamilton indicated "a minimum course of study, amounting to rather less than 200 pages (or parts of pages)," suitable for a first perusal, and he intended to have prepared a table containing references to this course. Such a table will be found at the end of the table of contents, but for the convenience of students of Physics, and of those desirous of obtaining a working knowledge of Hamilton's powerful engine of research, I have amplified it somewhat, duly noting, however, the minimum course.

I infer from the fragment of the author's preface that he proposed to sketch an outline of the method of exposition, of an elementary character and adapted to those readers to whom the subject is new. To those readers chiefly I address the following remarks:-

According to the plan of this work, whenever a new conception or notation is introduced, a series of illustrative examples immediately follows. Most of these involve no real difficulty, but occasionally a long and difficult investigation occurs even in the early parts of the book. Intricate investigations, which are merely illustrative, are everywhere omitted from the selected course.

The First Book deals with Vectors, considered without reference to angles or to rotations. In a word, it is concerned with the application of the signs +, -, and = to the algebra of vectors. The sign - is first introduced, and the sign + follows from the formula of relation $(b - a) + a = b$. Sections 3 and 4 (pp. 7-11) are occupied with a series of propositions concerning the commutative and associative laws of the addition of vectors, and the multiplication of vectors by scalars, or algebraical coefficients. Propositions such as these often appear to a student to be mere truisms, and unfortunately it is not easy to find elementary examples to convince him of the contrary. The addition of vector-arcs, he will find on p. 156, is not commutative, though it is associative. With the exception of a few passages noted in the table of a selected course, there is nothing in chaps. II. and III. essential to a good knowledge of the subject. They contain, however, an account of an extremely elegant theory of anharmonic coordinates, independent of any non-projective property, and intricate and powerful investigations of geometric nets and of systems of barycentres.

The Second Book treats of Quaternions considered as quotients of vectors, and as involving angular relations. It opens with a first conception of a quaternion as a quotient of two vectors, and thus the division of vectors is introduced before that of multiplication, just as in the First Book subtraction precedes addition. If $q = \beta : \alpha$ is the quotient of two vectors, $\beta$ and $\alpha$, it is natural to define the product $q. \alpha$ by the relation $q. \alpha = \beta$. It is soon found, if any vector $\gamma$ is selected in the plane of $\alpha$ and $\beta$, that the product $q. \gamma$ is a vector in the same plane whose length bears to that of $\beta$ the same ratio as the length of $\beta$ to that of $\alpha$, and which makes the same angle with $\gamma$ that $\beta$ makes with $\alpha$. Thus, from the first conception of a quaternion as a quantity expressing the relative length and direction of two given vectors, we have come to consider a quaternion as an operator on a special set of vectors, viz. those in its own plane. Observe that, so far, we have not arrived at the conception of the product of two vectors, nor of the product of a quaternion and an arbitrary vector. We have only reached the limited conception of the product $q . \gamma$ of a quaternion $q$ and a vector $\gamma$ in its plane, and while an interpretation is assigned to $q . \gamma$, as yet the product $\gamma . q$ is unknown.

After reviewing a class of quaternions derived by fixed laws from a given quaternion, a special class of quaternions, called versors or radial quotients, is considered in detail. The product of a pair of versors is found (p. 147) to depend on the order in which they are multiplied, that is $qq'$ is not generally equal to $q'q$ or the commutative law of algebraic multiplication is not true for versors, nor à fortiori for quaternions.

The multiplication of a special set of versors of a restricted kind occupies section 10, chap. I; and on p. 160 the famous formula

          $i^{2} = j^{2} = k^{2} = ijk = -1$ (A)

is deduced, in which $i, j$ and $k$ are right versors in three mutually perpendicular planes. This section contains the first example of a product of more than two versors, and it is shown that the multiplication of these specially related right versors is associative. Warned by the failure of the commutative law, it is necessary to determine if the remaining laws of algebra are valid in quaternions. In algebra, if we first form the product $bc$ and then multiply by $a$, we have the same result as if we multiplied $c$ by the product $ab$, and this associatiye law is expressed in symbols by the equation $a . bc == ab . c$. This is also true for quaternions, and it may be regarded as the chief feature which distinguishes quaternions from other systems of vector analysis. For example, Grassmann's multiplication is sometimes associative, but sometimes it is not. It is necessary to prove, moreover, that quaternion multiplication is distributive, or that $a(b + c) = ab + ac$. This is not true if $b$ and $c$ are vector arcs, even when $a$ is a number as shown on p. 156. Some of Hamilton's early investigations led him to a non-distributive system of multiplication in 1830.

Next a quaternion is decomposed in two ways:- (1) in section 11, into the product of its tensor and its versor; (2) in section 12, into the sum of its scalar and its right or vector part. This right or vector part, it is ultimately shown, may be identified with a vector; at present it is regarded as a right quaternion, or a quotient of two perpendicular vectors. By the first of these decompositions, "the multiplication of any two quaternions is reduced to the arithmetical operation of multiplying their tensors, and the geometrical operation of multiplying their versors"; and by the second the addition of quaternions is reduced to the algebraical addition of their scalar parts, and the geometrical addition of their vector part«. Thus it is proved (Arts. 206, 207) that the addition of the vector parts is reducible to the addition of vectors, and, as the addition both of scalars and of vectors is commutative and associative, so likewise is the addition of quaternions.

The multiplication of right quaternions, or of the vector parts of quaternions, is proved in Art. 211 to be distributive; and, as any quaternion is the sum of a scalar and a vector part, it is also proved that the general multiplication of quaternions is distributive. A long series of examples follows, some of which are not easy, including Hamilton's well-known construction of the ellipsoid.

Section 14 is entitled "On the reduction of the general Quaternion to the Standard Quadrinomial Form $(q = w + ix + jy + kz)$; with a First Proof of the Associative Principle of the Multiplication of Quaternions." This proof depends on the general Distributive Property lately proved, and on the Associative Property of the particular set of versors $i, j, k$ (Art. 161); but in chap. III. various proofs are given which are independent of these properties. The first proof is sufficient for all practical purposes.

The laws of combination of quaternions are now established. Addition (and subtraction) is associative and commutative; multiplication (and division) is associative and distributive, but not commutative.

Passing over the second and third chapters in this Second Book, which are chiefly complementary to the development of the theory, we find in chap. I., Book III., three lines of argument traced out in justification of the identification of the vector part of a quaternion with a vector. In fact a restriction is imposed, or a simplification is introduced, and this restriction or simplification is shown to be consistent with the results already obtained. In much the same way as a couple or an angular velocity is sometimes represented by a right line, a right quaternion and a vector of appropriate length, perpendicular to the plane of the quaternion, are now represented by the same symbol.

The scope of the remainder of this volume is, I think, sufficiently indicated in the table of contents. The foregoing sketch of the development of the calculus of Quaternions necessarily presents but a meagre view of the nature of this work; however, my object has been to carry out, as far as I could, the intention of its illustrious author expressed in the fragment of his preface.

Charles Jasper Joly
The Observatory, Dunsink
December 1898

2.1. Advertisement to the Second Edition.

I have reserved for the Appendix to this Volume the longer additional and illustrative notes which I have written for the new edition of the "Elements."

Some of those notes would have been inconveniently long as footnotes; others would have been inconveniently placed. For example, although the Note on Screws relates naturally to Art. 416 and that on the Kinematical Treatment of Curves to Art. 396, I have placed the Note on Screws before the Note on Corves because Hamilton's remarks on screw motion in the earlier Article required some development in order to make the Note on Curves easily intelligible. Accordingly the order of the notes has been arranged with reference to the notes themselves rather than with reference to the text. The selection and treatment of the subjects of these notes have been subordinated to the illustration of quaternion methods. I have not hesitated to sacrifice brevity for suggestiveness, and above all I have tried to render the notation as explicit as possible.

An analysis of the Appendix will be found on pages xlv-xlix.

For greater convenience I have provided an Index to the whole work referring to the pages, the volumes being distinguished by the numbers i and ii.

I take this opportunity of testifying to the extraordinary accuracy both of matter and of printing in the first edition of the "Elements." Every portion of the work bears evidence of Hamilton's unsparing pains. I cannot recall a single sentence ambiguous in its meaning, or a single case in which a difficulty is not honestly faced. I see no sign of diminished vigour or of relaxed care in those portions of the work written in his failing health. My task as editor has convinced me of the extreme caution with which any endeavour should be made to improve or modify the calculus of Quaternions.

In conclusion, I desire to express my thanks to the College Printer, Mr. George Weldrick, for the great care he has taken in printing this edition for the Board of Trinity College, and for his unvarying courtesy to myself.

Charles Jasper Joly
The Observatory, Dunsink
16 December 1900

3.1. Preface.

Readers of the Life of Sir William Rowan Hamilton will recollect that he undertook the publication of a book on quaternions to serve as an introduction to his great volume of Lectures. This Manuel of Quaternions was intended to occupy about 400 pages, but while the printing slowly progressed it grew to such a size that it came to be regarded by its author as a "book of reference" rather than as a textbook, and the title was accordingly changed to The Elements of Quaternions. By a curious series of events one of Hamilton's successors at the Observatory of Trinity College has felt himself obliged to endeavour to carry out to the best of his ability Hamilton's original intention. And on the centenary of Hamilton's birth a Manual of Quaternions is offered to the mathematical world.

Last year I was called upon by the Board of Trinity College to assist in the examination for Fellowship. I had long ago recognized that another work on quaternions was required, and this want was forcibly brought home to me by my new duties. A mathematician, whose time is limited, is frightened at the magnitude of Hamilton's bulky tomes, although a closer acquaintance with the Elements would reveal the admirable lucidity and the logical completeness of that wonderful book, and although the Lectures have a charm all their own. The student wants to attain, by the shortest. and simplest route, to a working knowledge of the calculus; he cannot be expected to undertake the study of quaternions in the hope of being rewarded by the beauty of the ideas and by the elegance of the analysis. And for his sake, though with reluctance I must confess, I have abandoned Hamilton's methods of establishing the laws of quaternions.

By a brilliant flash of genius Hamilton extended to vectors Euclid's conception of ratio. A quaternion is the mutual relation of two directed magnitudes with respect to quantity and direction as a ratio is the mutual relation of two undirected magnitudes with respect to quantity. From this enlarged view of a ratio, the calculus of quaternions is developed in the Elements. But the way is long and winding, and after much labour, I found I could not greatly shorten it or make it much less indirect. I therefore adopted another plan.

The two cardinal functions of two vectors are $S\alpha\beta$ and $V\alpha\beta$. These functions may be defined by the statements that $- S\alpha\beta$ is the product of the length of one vector into the projection of the other upon it, and that $V\alpha\beta$ is the vector which is perpendicular to $\alpha$ and to $\beta$, and which contains as many units of length as there are units of area in the parallelogram determined by $\alpha$ and $\beta$. Both these functions enjoy some of the properties of an algebraic product. They are distributive with respect to each of the vectors.

The product of the vector $\alpha$ into $\beta$ may be defined to be the sum of these functions,

          $\alpha\beta = S\alpha\beta + V\alpha\beta$.

This is a quaternion - the sum of a scalar and a vector. A product of a pair of vectors is distributive but not commutative. It is now necessary to define the product of a quaternion $(q)$ into a vector $(\gamma)$, and we say that it is the sum of the product of the scalar (Sq) into $\gamma$ and the product of the vector $(Vq)$ into $\gamma$, or that

          $q . \gamma = Sq . \gamma + Vq . \gamma$.

From these principles it follows almost immediately that quaternion multiplication is associative as well ail distributive.

Division is seen to be deducible from multiplication, and on p. 12 we arrive at the important result that every function of quaternions formed by ordinary algebraic processes is a quaternion, scalars and vectors being considered to be special cases.

What we may call the grammar of the subject may be said to terminate on p. 20, the law of combination of quaternions having been established, the five special symbols $S, V, K, T$ and $U$ having been defined and their chief properties explained, various constructions for products and quotients having been made, and the non-commutative property of multiplication having been illustrated by conical rotations and otherwise.

In the succeeding chapters, I have not scrupled to introduce, either in the articles in small type or in the worked examples in small type, illustrations of the applications of quaternions to subjects that can hardly be supposed to be familiar to the beginner in mathematics. It is suggested in the table of contents that these more difficult portions should be omitted by a beginner at first reading. The book is, however, primarily intended for those who commence the study of quaternions with a fair knowledge of other branches of mathematics; in other words, it is written for the majority of those at present likely to read quaternions because, as yet, the subject is not generally taught in elementary classes. On the other hand, I have abstained from printing examples of an artificial nature, and I have avoided unnecessary difficulties.

Although this book may be regarded as introductory to the works of Hamilton, it may also to some extent be considered as supplementing them. Many of the results contained in it have appeared only in the publications of learned societies, and many others are believed to be novel, It is possible, therefore, that this volume may be found to have some points of interest for the advanced student of quaternions. He will find, for example, that quaternions lend themselves to the treatment of projective geometry quite as readily as to investigations in mathematical physics and in metrical geometry.

By means of a somewhat elaborate table of contents, modelled on those prefixed by Hamilton to his Lectures and Elements, and by the aid of a full index and numerous cross references, I trust that the contents of this book will be found to be fairly accessible to the casual reader as well as to the systematic student. It must be remembered, however, that the objects of a work of this nature are to introduce a subject of the highest educational value, and to develop a powerful and comprehensive calculus. Such ends can be attained only by illustration and by suggestion, and it is not easy to tabulate methods of investigation.

It would be impossible to overestimate what I owe to Hamilton's Lectures on Quaternions (Dublin, 1853) and to his Elements of Quaternions (London, ]866, 2nd edition, in two volumes, with notes and appendices by C J Joly, London, 1899, 1901). The admirable Elementary Treatise con Quaternions (3rd edition, Cambridge, 1890), by the late Professor P G Tait - who has done so much for quaternions by his classical applications of Hamilton's operator ∇ - has also been very useful. Other writers to whom I am indebted are referred to in the text. I am glad to have this opportunity of offering my thanks to my respected friend, Benjamin Williamson, Esq., F.R.S., Senior Fellow of Trinity College, Dublin, for his great kindness in assisting me with a considerable portion of the proofs. I am also indebted to him for the uninterrupted encouragement he has given me, alike privately and in his official capacity as a member of the governing body of Trinity College, in my attempts to render Hamilton's work more widely known.

Charles Jasper Joly
The Observatory
Dunsink, Co. Dublin
1 January 1905

3.2. C G Knott, Review: Manual of Quaternions, by C J Joly, The Mathematical Gazette 3 (53) (1905), 229-231.

Tait has said somewhere that Hamilton first invented the quaternion and then discovered it. The invention consisted in the conception of the three imaginaries $i, j, k$, with their special laws of combination, and in the construction of an associative algebra with four fundamental units. The quaternion was then discovered to be a complex number representing the ratio of two vectors or directed lines in space. Hamilton was the first who clearly recognised the value of the associative law; and in spite of many imitations his system remains the only tridimensional system of vector analysis governed by this law.

The mathematical world has, however, been slow to recognise the essential merits of Hamilton's calculus. A vast deal of ingenuity and time has been spent by some in finding new notations of the quantities and operators peculiar to quaternions; some have even been beguiled by these notations into a rediscovery of theorems as old as Hamilton's Lectures. The curious thing is that it does not appear that any really fresh mathematical truth has been brought to light by the inventors and users of vector notations intended to supersede Hamilton's original system. Here, of course, we refer only to tridimensional applications.

It is very refreshing then to open the pages of a book whose author, boldly accepting the form of the calculus as Hamilton developed it, proceeds to unfold its beauties and strength with all the skill of a practised hand. The book begins with a very brief chapter on the addition and subtraction of vectors, a part which necessarily occupied a considerable section of Hamilton's and Tait's treatises. The vector conception has now crept into our elementary books, and in due course will probably become a conspicuous feature even of our most elementary geometries. Graphical methods have within recent years transformed our teaching of algebra; and the vector as a geometrical entity is essentially graphical.

Professor Joly wisely assumes that for students who have made some progress in mathematics the law of vector addition calls for little elaboration. He devotes five pages to it and then plunges into quaternions proper. He takes what he believes to be the "shortest and simplest route." The student he says "cannot be expected to undertake the study of quaternions in the hope of being rewarded by the beauty of the ideas and by the elegance of the analysis. And for his sake, though with reluctance I must confess, I have abandoned Hamilton's methods of establishing the laws of quaternions." Perhaps so, but it is a poor student who despises logical development, beauty of idea, elegance of method, and is content with a "working knowledge of the calculus." We confess to an uneasy feeling that principle has here been sacrificed to expediency. There is a suggestion of "tumbling over the wall" and not coming "in at the Gate which standeth at the beginning of the way." We have often wondered how many students study Clifford's Dynamic for the sake of learning dynamics. Possibly none; and probably as few have learned quaternions from that book. Although there is some initial similarity between Professor Joly's method and the tentative nibbling at quaternions which characterises Clifford's book, there is almost immediately a vast divergence. The true quaternion is introduced on page 9, and it dominates the whole treatise. This is as it ought to be. It is possible, as Heaviside has shown, to use effectively much of the notation of quaternions without explicit use of the quaternion itself; but sooner or later the lack of it will be felt. The student should never lose sight of the fact that $S\alpha\beta$ and $V\alpha\beta$ together form by addition a quantity which, however it may operate on or be taken in conjunction with a like quantity, gives rise to a quantity of the same analytical nature. This is the central doctrine of the quaternion calculus.

To give any complete idea of all that Professor Joly's volume contains would be practically to reproduce his table of contents. Beginning with the simpler applications to trigonometry and to the geometry of plane and sphere, he quickly passes into the peculiarities of quaternion differentiation and into the exquisite theory of the linear vector function (or matrix), after which he is ready for all kinds of applications in geometry of curves and surfaces, and in kinematics and dynamics. The reader cannot fail to be impressed with the directness of the method in all these applications, especially if he is familiar with the ordinary modes of attack. In virtue of this directness of attack and the extraordinary conciseness of notation more detail can be packed into one quaternion page than into three or four pages of ordinary analysis. By what other method, for example, could systematic discussions of line, surface and volume integrals, of spherical harmonics, heterogeneous strain, elastic vibrations, and electro-magnetic theory be given in less than fifty pages? In the variety of the mathematical and physical subjects taken up there are only two other books which can compare with Professor Joly's Manual, and these are Hamilton's Elements and Tait's Treatise.

The greater part of the book is necessarily a development of much that is to be found in the pages of Hamilton, Tait, and McAulay; but Professor Joly has a characteristic style of his own, more nearly akin to Hamilton's than to Tait's. In the last two chapters especially are the author's additions more in evidence. These are on Projective Geometry and Hyperspace. The former is based upon a new interpretation of the quaternion; and in the latter Professor Joly gives a sketch of the properties of associative algebras applicable to n-dimensional space.

Professor Joly has certainly succeeded in his aim of providing the student with a working book. He takes excursions into many fields of mathematics pure and applied, and the treatment is not superficial. Important applications are worked out in detail; and numerous examples are given by which the student may test his progress. Let the reader accept on trust the initial assumptions and developments, and work earnestly through the succeeding chapters. He will come out in the end a practised quaternionist.

C G Knott

The perusal of these hundred quarto pages suggests a consideration of the present state of quaternion theory and the peculiar aspects with which it is regarded by different classes of mathematicians. It is difficult to say which is more deplorable, the total ignorance of some, or the disproportionate enthusiasm of others. It is surely possible to admire the elegance and brevity with which, in particular, certain kinematical theorems can be expressed, to appreciate the isotropy on which Hamilton laid so much stress, and at the same time to recognise that the special nature of quaternions and the clumsiness of much of the notation connected with them impose stringent limitations on their useful application. Quaternions should be regarded initially from the algebraic standpoint; the geometrical properties easily follow as the interpretation of the results of permissible algebraic processes. In this view, quaternions occupy a small niche in the general theory of complex numbers; they are particularised by the integer four and by the special relations among the complex units, and hence must be necessarily of limited application and lead to results capable of but slight generalisation. From the utilitarian point of view quaternions have little to justify their extensive use. Except in few special problems, nothing is gained by their use that is not as conveniently and effectively attained by the more natural and more general notation of matrices. Both notations aim at dealing with many quantities at once, but in the quaternion notation these quantities are bound together with chains which have to be broken by the introduction of clumsy adventitious symbols, whereas in the matrix notation there is nothing superfluous. For instance, $\Sigma p_{s}q_{s}$ is in matrix notation simply $pq$, but when $p$ and $q$ are quaternions we have to multiply one by the conjugate of the other, and then pick out the scalar part of the product, writing it $Spq'$ or $Sp'q$.

The memoir under review falls roughly into two parts; the first half gives the theory of the four-rowed square matrix in the language of quaternions. We are introduced to the linear quaternion function, which has the same effect as the general linear substitution in four variables. If $A$ is the matrix of the coefficients in this substitution, and $A'$ is its conjugate, then obviously $A + A'$ is symmetric, and $A - A'$ is skew. Thus we are led to associate a quadric surface and a linear complex, which, with the invariant tetrahedron of the substitution, form the materials for a considerable amount of elementary geometry.
...

The latter part of the memoir is devoted to the bilinear quaternion function, which is a linear combination, with quaternion coefficients, of four ordinary bilinear forms in two sets of variables. These complicated expressions are manipulated with wonderful skill, and made to yield a vast number of important geometrical results. A particularly interesting application is to the general quadratic transformation of space, in which any plane is transformed into a Steiner's quartic surface.