Rece. 1-16-41. H.R. J. PREFACE. THOUGH the subjects treated of in this volume, have, individually, and more especially in the relations which subsist between them, engaged my attention more frequently, more deeply, and I may add more delightfully, than any other subjects of a scientific nature, which I have made the object of thought; and though very many years have elapsed since I first felt the want and the desire of possessing some such book—and even since I came to the resolution of attempting its production, and had in some sort sketched its plan-yet, I fear, and indeed feel, that the execution of it stands more in need of a preface, or explanation, or apology, than any work which I have hitherto attempted. I am aware that it is not a book for even the moderately learned in systematic mathematics, far less for those whose talents and acquirements do honour to the science and extend its boundaries. I am somewhat apprehensive, too, that it may not prove to be exactly the book which is to afford to the young and the unlearned the spirit of mathematical science, and the force of mathematical truth-the communicating, or, to speak more correctly, the inspiring, or rather the stimulating of which, is the main object by which I have been guided both in preparing and in executing the work. My chief ground of apprehension on this latter point is the fact of there being no similar book, by the success of which I could be encouraged, or by the failure of which I could be warned. Thus I have been thrown upon my own conception of what was most likely to be useful, without any direct experience on the part of others, by which I could be assisted or guided. But though I laboured under this, perhaps the greatest disadvantage that an author could have had, I feel that I also enjoyed some peculiar advantages. My notions on the subjects, and on the mode of conveying information respecting them, were originally my own. A disciple of no school, and a follower of no master, I had no mannerism of others to come between me and the truth: and it so happened that almost as fast as I could acquire some knowledge, not of a whole subject, but of the successive parts of it, I had the advantage of trying the effect of what I had acquired, and how I had acquired it, in the instructing of others; and very frequently I found that the indirect lesson which I derived, from the effect produced upon the student, was of even greater advantage to me than what I did for myself, and probably greater than I could have de rived even from able instructors. I need not say that these few circumstances are not mentioned in any spirit of boasting or self-gratulation, for truly there is not in them anything of which even the vainest man could boast. Besides, after a man has been hammered, pretty smartly and pretty constantly, upon the world's anvil for half a century, though the metal of his mind may not thereby be changed, yet it is beaten to so much compactness, that there are few pores in it for holding so unsubstantial a thing as vanity. I have mentioned them rather, as unavoidable reasons why this book should be so different from the ordinary books of elementary mathematics; in addition to which I have endeavoured to supply what none of those books singly, or perhaps the whole of them taken together, can supply. In saying this I do not mean that there are many new truths in the volume, or that there is one known truth stated more clearly than it is to be met with elsewhere. But considering the vast number of such books which it is necessary to study, with profound and patient attention, in order to get possession of all the truths which are necessary for having a tolerable knowledge of even the first elements of mathematics, in the three departments of General Quantity, or ALGEBRA ; Numbers, or ARITHMETIC; and Magnitudes, or GEOMETRY; and the portion of life that even this, which after all is only a sort of mechanical labour, must consume, before the student is in a condition for beginning to generalise; it is easy to see that the business or the pleasures of the world, must neccssarily take hold of a vast majority, even of students, before they have arrived at the commencement of this, the truly mental and most useful part of Geometry. Having felt this very severely in my own case-and there is too much of the bitterness of regret mingled with it to allow me to forget it-I have endeavoured to start with generalisation at the very outset of this volume, and to hold fast by it on every occasion, regardless how much it might break in upon the symmetry of the book, or the smoothness of its execution. Such being the case, this work is not to be considered as a book of reference, from which particular truths, or formulæ for the solution of particular problems, are to be borrowed, without reasoning, and often I may add without instruction; neither is it a task-book, to be conned by rote in successive fragments, and parroted without knowledge, until active employment of the mind cause them to be forgotten. It is strictly, (that is to say in so far as I can judge of it, destitute as I am of an external standard of judgment,) what its title expresses—“ POPULAR MATHEMATICS;" that is to say, a book which is meant to be read through, and which is intended to inspire those who, from too tender age or want of opportunities and means, have not acquired a knowledge of mathematical science, with a general perception of its nature, a feeling of its power as an instrument both of wisdom and of working, and the love of a farther acquaintance with it. Every one who has caught even one little ray of the glorious light of this science, must feel that it is as powerful as it is brilliant ; |