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 necessarily

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 ;

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,

and I shall abandon my argument, and join ever after "the herd of gentlemen who write with ease."

So much for the plan and purpose of the work; and the execution can be best seen and judged of in the work itself; therefore I shall only state further that I have been careful to bring forward the three branches in such an order of succession, as that the reader who reads for instruction, (as I sincerely hope many will,) may call them all to his aid whenever he feels it necessary. I have dwelt longest upon those general points which appeared to me to possess in the highest degree the two qualities of furnishing the greatest number of inferential truths and stimulating the reader to seek out those truths; and I have been more anxious to create a love of the science, than to carry the particular departments of it to a great extent. To use a homely simile, if a man gets lamed before he commences a journey, it is far better to cure him and let him start in his own strength, than to carry him half way and leave him in his lameness. But this simile, homely though it is, applies to every branch of education, and to mathematics in an especial manner. To talk about teaching a person a science, is like talking about a lame man's performing a journey when he is carried; but, if we can succeed in awaking the desire and arousing the capacity, the party will learn, not only without our teaching, but in spite of our opposition; and this is the grand object which should be aimed at by every well-wisher to the mental and moral character of the human race.

I cannot say that I shall conclude this preface-for the

same train of thought is continued in the introductionbut I shall conclude the present writing by claiming the suffrages of the public in favour of my purpose, how much soever they may blame the execution of it,-only adding, that if the present volume shall meet with a reception at all proportionate to the labour it has cost me, I purpose following it up by another, carrying the three branches of the science as far as they are required by those who are not professional mathematicians.

Grove Cottage, Chelsea,

July, 1836.

ROBERT MUDIE.