the writing of the "ELEMENTS" had cost him-labour both mental and mechanical; as, besides a mass of subsidiary and unprinted calculations, he wrote out all the manuscript, and corrected the proof sheets, without assistance. And here I must gratefully acknowledge the generous act of the Board of Trinity College, Dublin, in relieving us of the remaining pecuniary liability, and thus incurring the main expense, of the publication of this volume. The announcement of their intention to do so, gratifying as it was, surprised me the less, when I remembered that they had, after the publication of my father's former book, "Lectures on Quaternions," defrayed its entire cost; an extension of their liberality beyond what was recorded by him at the end of his Preface to the "Lectures," which doubtless he would have acknowledged, had he lived to complete the Preface of the "ELEMENTS.' He intended also, I know, to express his sense of the care bestowed upon the typographical correctness of this volume by Mr. M. H. Gill of the University Press, and upon the delineation of the figures by the Engraver, Mr. Oldham. I annex the commencement of a Preface, left in manuscript by my father, and which he might possibly have modified or rewritten. Believing that I have thus best fulfilled my part as trustee of the unpublished "ELEMENTS," I now place them in the hands of the scientific public. PREFACE.* [1.] THE volume now submitted to the public is founded on the same principles as the "LECTURES," which were published on the same subject about ten years ago: but the plan adopted is entirely new, and the present work can in no sense be considered as a second edition of that former one. The Table of Contents, by collecting into one view the headings of the various Chapters and Sections, may suffice to give, to readers already acquainted with the subject, a notion of the course pursued: but it seems proper to offer here a few introductory remarks, especially as regards the method of exposition, which it has been thought convenient on this occasion to adopt. [2.] The present treatise is divided into Three Books, each designed to develope one guiding conception or view, and to illustrate it by a sufficient but not excessive number of examples or applications. The First Book relates to the Conception of a Vector, considered as a directed right line, in space of three dimensions. The Second Book introduces a First Conception of a Quaternion, considered as the Quotient of two such Vectors. And the Third Book treats of Products and Powers of Vectors, regarded as constituting a Second Principal Form of the Conception of Quaternions in Geometry. * This fragment, by the Author, was found in one of his manuscript books by the Editor. TABLE OF CONTENTS. BOOK I. ON VECTORS, CONSIDERED WITHOUT REFERENCE TO CHAPTER I. FUNDAMENTAL PRINCIPLES RESPECTING VECTORS, Pages. 1-102 SECTION 2.-On Differences and Sums of Vectors, taken two SECTION 1.-On the Conception of a Vector; and on Equality of Vectors, 1-3 by two,. 3-5 5-7 8-11 SECTION 3.-On Sums of Three or more Vectors, This short First Chapter should be read with care by a beginner; any misconception of the meaning of the word "Vector" being fatal to progress in the Quaternions. The Chapter contains explanations also of the connected, but not all equally important, words or phrases, "revector," "provector," "transvector," "actual and null vectors," "opposite and successive vectors," "origin and term of a vector," "equal and unequal vectors," "addition and subtraction of vectors," "multiples and fractions of vectors," &c.; with the notation B-A, for the Vector (or directed right line) AB: and a deduction of the result, essential but not peculiar‡ to quaternions, that (what is here called) the vector-sum, of two co-initial sides of a parallelogram, is the intermediate and co-initial diagonal. The term "Scalar" is also introduced, in connexion with coefficients of vectors. * This Chapter may be referred to, as I. i.; the next as I. ii.; the first Chap. ter of the Second Book, as II. i.; and similarly for the rest. †This Section may be referred to, as I. i. 1; the next, as I. i. 2; the sixth Section of the second Chapter of the Third Book, as III. ii. 6; and so on. Compare the second Note to page 203. b CHAPTER II. Pages. APPLICATIONS TO POINTS AND LINES IN A GIVEN PLANE, 11-49 SECTION 1.-On Linear Equations connecting two Co-initial Vectors,. 11-12 SECTION 2.-On Linear Equations between three Co-initial After reading these two first Sections of the second Chapter, and perhaps the three first Articles (31-33, pages 20-23) of the following Section, a student to whom the subject is new may find it convenient to pass at once, in his first perusal, to the third Chapter of the present Book; and to read only the two first Articles (62, 63, pages 49-51) of the first Section of that Chapter, respecting Vectors in Space, before proceeding to the Second Book (pages 103, &c.), which treats of Quaternions as Quotients of Vectors. 12-20 SECTION 3.-On Plane Geometrical Nets, 20-24 24-32 32-35 Among other results of this Chapter, a theorem is given in page 43, which seems to offer a new geometrical generation of (plane or spherical) curves of the third order. The anharmonic co-ordinates and equations employed, for the plane and for space, were suggested to the writer by some of his own vector forms; but their geometrical interpretations are assigned. The geometrical nets were first discussed by Professor Möbius, in his Barycentric Calculus (Note B), but they are treated in the present work by an entirely new analysis: and, at least for space, their theory has been thereby much extended in the Chapter to which we next proceed. CHAPTER III. APPLICATIONS OF VECTORS TO SPACE, SECTION 1.-On Linear Equations between Vectors not Complanar, . 35-49 49-102 It has already been recommended to the student to read the first two Articles of this Section, even in his first perusal of the Volume; and then to pass to the Second Book. 49-56 SECTION 2.-On Quinary Symbols for Points and Planes in 57-62 |