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and that when we come to work our way to the knowledge of things around us, from the sod on which we stand, to the most remote luminary in the heavens upon which two hundred millions of miles will tell as a measurable fraction, will readily admit, that mathematics is not only the line wherewithal to measure, and the balance wherein to weigh, but that it is the wedge to cleave asunder whatever is too gnarly and stubborn, and the lever to heave aside whatever is too weighty for the other apparatus of thinking and executing.
Those who have formed their notions from those nominal mathematicians, who idle with the disjointed bones of the science in the absence of the life, are apt to suppose, and sometimes to say, that mathematical science has a tendency to curb the fancy, and pedantify the mind. Among all the blunders of ignorance there is not one more gross than this; and we might appeal with triumph to mathematicians of every age as leaving recorded in their writings, abundant evidence of the most exalted and expanded imagination, and the most chaste and lively fancy. I shall mention only one or two names ; and these of the last and the present generation. Who in his time excelled or even equalled the late John Playfair of Edinburgh, (with whom I have again and again discussed the subject and plan of this work,) in power, in purity, and in beauty of style? And who, in our own times, writes like Whewell or Herschel? Find me the unmathematical man that shall set an idea before the mind, as a mental and tangible solid, with the same power and truth as either of them,
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