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It is the intention of the following introductory work to exhibit, in moderate compass, a tolerably comprehensive development of the fundamental principles of algebraical science. In a former publication I have extended these principles to the more advanced parts of the subject—to the general investigation of the Binomial Theorem; to the doctrine of Series; to the elements of Probabilities, &c. But it is not easy, in a single duodecimo volume, to prepare the way to this higher class of inquiries with that amplitude of detail, and fulness of practical illustration, which beginners sometimes need, in order to give them the necessary tact and expertness in what may be called the more delicate manipulations of algebra. I think the present Introduction will be found to supply all that is requisite in this preparatory business, and to lay a sufficient foundation for an extended course of analytical study. It is a task that has been urged upon me by others-persons of enlarged professional experience, and who have had much more to do than I have with the intellectual training of youth in the earlier years of life. I still think, however, that my former book will sufficiently answer the purposes of minds a little more mature; that is, if those who, having acquired in some degree the important habit of thinking for themselves, stand less in need of examples to confirm and illustrate precepts; and in whom the character of the assiduous student has succeeded to that of the volatile schoolboy.

It is principally for schoolboys that this introductory work is written; and accordingly it abounds with examples, which, by detaining the precipitate, and occupying and stimulating the sluggish, give time and opportunity for both classes fully to possess themselves of every general principle exemplified before advancing to anything new. In these practical illustrations of the theory, I have chosen to err, if at all, more on the side of excess than of defect: examples can more easily be expunged than supplied. But I should be among the last to countenance the practice, sometimes adopted in elementary books of science, of overloading the work with examples at the expense of the theory. I do not know a plan more to be reprobated in books of this kind, than the making of them mere depositories of rules and examples: such books are unworthy to be called introductions to science, and are as derogatory to the writers as to their subjects. If schoolboys are incompetent to appreciate a simple process of reasoning—if they cannot be made to understand a logical deduction-mathematics should be kept from schoolboys. But it appears to me that a writer is greatly to blame to proceed upon any such assumption, and so denude his instructions of nearly all that can render them valuable or give them a claim to the title they bear. It is his business faithfully to discharge his duty to his science and to himself, without descending to any degrading compromise with the supposed incapacity of his readers. The writer of an elementary book should aim at something more than the mere harmless occupation of a few schoolboy months: he should aspire to the higher object of imbuing the young mind with the sound principles of accurate science, animated by the hope that he may, perhaps, be instrumental in awakening dormant energies, and in giving an early, but at the same time, a healthy impulse to youthful talent.

It is only in this way that “ an elementary writer"-—often a disparaging designation—can cease to be contemptible, and can hope to render any essential service to the cause of science; and it is the only way in which he can secure the abiding good will—it may be the grateful remembrance, of those whose first steps in its pursuit he undertakes to direct. These are results that should be highly prized by those who thus devote their lives to the arduous labours of instruction, as they but too often constitute their only reward. The humblest book on science should perform its part in preparing the way towards the profoundest researches: it may be studied with one object by one, and with a different object by another; but whether the reader choose to content himself with the merest rudiments, or aspire to a future familiarity with the writings of Newton and Laplace, are alternatives with which the elementary writer has nothing to do: within the limits of his undertaking his business is so to unfold the principles of his subject as to preclude the reproach of having either suppressed the truth or taught that which, hereafter, may be required to be unlearned.

I have made these remarks lest it should be hastily supposed that this little book-in some degree subordinate in its aim to the volume already adverted to, is intentionally deficient in that theoretical accuracy which is always expected to characterise works of higher pretensions. If a looseness be observable in the enunciation of any principle delivered in it, or a want of scientific precision in the exposition of any truth or doctrine, it must be attributed to unintentional oversight, and not to the circumstance that I was writing for mere schoolboys--a fact which has made me only the more anxious to avoid blemishes of this kind.

Without anticipating the annexed table, by entering here into any sum

mary of the contents of the volume, I may mention that some things have been introduced which, to a fastidious critic, may seem of questionable propriety in a book designed for beginners: I allude to the topics discussed in the last three chapters of the work.

One of these chapters is devoted to a brief exposition of the Numerical Solution of Equations beyond the second degree; and a sketch of Horner's process of development is exhibited in a form I think sufficiently intelligible to the young algebraist. In connection with this process there is given a new method for the discovery of the leading figure of a root—a preliminary absolutely indispensable to the successful application of that important principle. This method, as I have elsewhere shown, is very useful in the analysis of equations in general, and considerably facilitates the practice of Budan's theorem for that purpose. I formerly thought that researches in reference to this peculiar, and somewhat extensive, department of algebra, should be kept distinct from the other portions of the subject, and be removed a little out of the ordinary routine of school-mathematics. But I have seen reason to change this opinion. Reflections upon the neglect which Horner's valuable improvements have met with in quarters where one would expect that every discovery in science would be cordially recognised, and from which it would be widely disseminated, coupled with the fact that this great step in one of the most practically important branches of algebra, has been fully appreciated and adopted in instruction almost exclusively by men unconnected with our universities---private teachers and ordinary schoolmasters—have convinced me that things of this kind can be saved from oblivion only by being incorporated in our elementary manuals of instruction. It is true that one very distinguished ornament of the Cambridge University, Professor Peacock, took the earliest opportunity to express his favourable estimate of Horner's labours; and there is no doubt that when the masterly work on algebra by that author is completed, the University will be, in what may then be deemed, legitimate possession of these yaluable improvements in algebra, which, although made by an humble individual, will be sufficiently enhanced by the authority which recommends them, to render it no longer derogatory to that learned society to embody them in their system of education. The writings of Professor De Morgan, of the venerable James Lockhart, of Dr. Rutherford, and, above all, of my lamented friend, the late Professor Davies, have done much to give currency to Horner's researches; but after the lapse of more than thirty years since its first publication, Horner's method of approximating to the real roots of a numerical equation ought to be as familiar to the young algebraist as the operation for the square root is to the arithmetician.

The other topic, which some may consider a little out of place in an introductory book of this kind, is that on the Theory of Logarithms, with which the volume closes. I have touched upon this subject chiefly to

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