THE Mathematical Sciences are at once the most perfect and the most valuable portion of human knowledge. They are valuable, not only for their own direct applications to the business of life, for the number of sciences to which they are the key, and the number of arts of which they are the foundation, but they have a value, higher and more important, if possible, in the strength which they give to the mind, and the exercise which they afford to its noblest faculties.

There is, indeed, no road to clear, forcible, and connected reasoning, but that which is opened up by the Mathematician; and whatever be a man's profession or station in the world, we are always able, from his mode of stating a proposition, or conducting an argument, to say whether he be or be not a Mathematician. In every other department of knowledge, there is some uncertainty-some hypothesis assumed, of which the foundation is unsearched or inscrutable, or something which hinges upon the undefined and undefinable properties of mind or of matter, or upon the contingency of events. Is language the medium through which one would arrive at logical precision? Then living language is mutable, and dead language is ambiguous. Is it human nature? Then who can gauge the mind, who can number its propensities, or measure its eccentricities? Is it life and manners? Then the modes vary with every individual and with every hour. Is it medicine? Then the life of that man is short indeed who has not seen fifty nostrums and modes of treatment rise into vogue and sink into neglect, without any reason assigned, or assignable, either for their rise or their fall. Is it law? Then who shall numbe

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the absurdities of the statute-book, to say nothing about the practice? Chemistry is one tissue of puzzles and disputes; and in pure Physics, it is not much betIn the physical sciences there is indeed a mathematical element; and so far as this goes, they are clear, orderly, and precise; but the moment that it is left out we are in the regions of theory-the wilds of doubt and uncertainty. With the student of most of these, the ground is unstable under his feet, the path by which he advanced has closed upon him, and that by which he has still to advance has not opened: with the mathematician every step is upon sure ground; and he sees how he came and where he is to proceed. Nothing that is contingent or ideal is admitted. He hunts out every lurking sophism, rejects the smallest aberration from truth, and goes on with absolute confidence and absolute certainty.

A system of mental discipline, so valuable for the extensive and positive advantages which it affords, cannot be pressed too earnestly upon the attention of the public, or come before them in too many shapes; and it is to be regretted, that in this age of pretended improvement, the soundness and certainty of mathematical cultivation should have given place to something far more light and flimsy-to the mere blandishments of the Belles Lettres, or such simple preparation for a man's individual trade, as may make him a handy tool for carrying on that trade, but nothing else. In proof of this, let any look into the works of those called (not the Learned, according to the good old English phrase) but the Literati, and he will find, amid all the sparkle of their abundance of words, and all their affectation of sentiment, there is a plenteous lack of depth of thought and soundness of principle.

Let any one turn to the Magazines, those reputed barometers of taste, and he will find that, with one exception, the Monthly Magazine, there is not in any of them a single scrap of science, or a single article aiming at a higher or more durable character, than the idle amusement of an idle hour. Chapters of novels and tales, which have been rejected in the shape of volumes; rhymes, in which there is neither poetry nor point, and slender critiques upon matters still more slender, have come in the place of the profound remarks and masterly dissertations of the Johnsons, Hawksworths, Prices, Priestleys, and Thomsons, who once gave vigour and value to our periodical literature. In the regular books it is not much better; we have tales and travels, accounts of ideal beings who never could exist, and of real ones who will not be remembered; and even in the scientific departments, we have the science of skulls and of shells, instead of the dissection of mind, and the display of the grandeur of nature. The whole structure has, in short, become light and frivolous; and though there be no "royal road to Geometry" now, any more than there was in the time of the Grecian sage, we have found out a bye-path to Fame, such as it is, by which Geometry may be avoided. Under these circumstances, every attempt to bring genuine science within the range of a greater number of readers is praiseworthy.

In the following pages, it has been the object of the compiler to reject every thing trivial, and to present a body of mathematical and physical knowledge as substantial and as complete as it is unobtrusive. He has been sedulous to exclude every thing merely ornamental; and by doing this he hopes that he has squeezed into this little volume all the really useful

matter which is to be found in works of far higher price and loftier pretensions. Science is not a thing of fashion, and therefore little novelty can be expected. Discoveries and inventions in pure Mathematics were never abundant at any period, and latterly they have been exceedingly few.

As the compiler has himself had a good deal of practice in communicating mathematical knowledge, he hopes he may be pardoned if he offers a few words on what to him appears to be the most successful way of pursuing the study of them. Number, and form, and quantity, are the three subjects of all mathematical speculation. Number is the most simple, perhaps the most simple of all subjects of correct and rational enquiry; and therefore it is the one with which to begin. The notation, and common operations are easily understood; and these being once managed, a most beautiful field of speculation is opened up. The doctrine of prime and composite numbers, the doctrines of divisors, of fractions, and of ratios, and the doctrine of roots and powers, are all beautiful in themselves, and prepare the student for entering upon the more general subject of Algebra. That science, as far as relates merely to numbers, should immediately follow the study of Arithmetic; and then the mind is prepared for the elements of Geometry. In the study of these, old Euclid never has been, and probably never can be, equalled; and so the first book of his elements may follow the study of the notation, the common rules, and simple equations in Algebra. Before entering upon the second book of the elements, the student may proceed a little way with quadratics, and these and the second book will mutually assist each other.

When quadratics are well understood, the whole re

maining books of the elements both of Geometry and of Trigonometry are easily mastered; and then may come the higher equations. After this, the elements of the fluxionary or differential calculus may be studied; and then the doctrine of curves; and the more abstruse parts of the calculus. And, when this has been done, the student is ready for the mixed sciences; and, with such preparation, his progress can hardly fail in being both sure and rapid. Before, however, he proceeds too far in the mechanical sciences, it is advisable that he study the elements of Chemistry, just for the purpose of enabling him to decide what phenomena are mechanical and what are not. Such a course of study as we have been sketching, would, if rightly gone about, not occupy more time than most boys waste upon a few idle, or at least superficial, accomplishments, and it would fill society with men of another description than are now found in them.

To contribute to such a result is one of the objects of this Dictionary; and we shall close this preface by mentioning the names of a few of the books which would be found useful as auxiliaries, or which would furnish those details which cannot be looked for in a work so small and so condensed as this:-For a general synopsis, perhaps no book is better than Nicholson; but Bezout, and some others of the French authors, are more systematic. Of Arithmetic, the books are many; but in English few of them are profound. Barlow's Theory of Numbers is among the best. The practical ones are without number, and many of them without variety. Joyce, Hutton, Bonnycastle, Walkingham, Vyse, and far from the worst, Dr. Hamilton, may be noted. In Algebra, Euler is at once the