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reading and inquiries, and increasing the pains it endeavored to avoid.”
This, though it may seem to concern none but direct scholars, is of so great moment for the right ordering of their education and studies, that I hope I shall not be blamed for inserting of it here, especially if it be considered, that it may be of use to gentlemen too, when at any time they have a mind to go deeper than the surface, and get themselves a solid, satisfactory, and masterly insight in any part of learning.
[We have given in the foregoing article, the judgment of three scholars and philosophers whose reputation gives much weight to their opinions, on the subject of classical education. The plans and methods proposed by them, with that recommended in the extracts from Prof. Lewis in the two last numbers of the Annals, are the most important that have been suggested. Our readers are competent to form a decision on their respective merits. It ought however to be remembered that the plans of Ascham, of Millon, and of Locke were designed to attain ends somewhat diverse, and adapted to different classes of learners. That of Ascham is suited to private tuition, and accurate knowledge,—that of Milton will prepare the undertaker of it for extensive and thorough scholarship,—while Locke's is fitted only for a “gentleman” whose classical studies are a mere matter of accomplishment, and whose object can be reached by very superficial acquirements. Experience has we think fully shown that Locke's plan is almost useless for all the purposes of true knowledge, and of doubtful value as a means of discipline; while that of Ascham, the excellence of which lies in its difference from Locke's, and not, as often supposed, in its identity, commends itself more thoroughly on the more thorough trial. We think also that the aid the student may derive from an exact knowledge of the rules of syntax has not been fully appreciated by either Milton or Locke, and that their error was owing rather to a partial purpose, or a peculiar experience, than to a careful consideration of the true value of them.]
Art. II.-TREATISES ON ALGEBRA.
BAILEY'S ALGEBRA. First Lessons in Algebra, being an Easy Introduction to that science ; designed for the use of academies and common schools. By Ebenezer Bailey, Principal of the Young Ladies' High School, Boston.
Davies' ALGEBRA. First Lessons in Algebra, embracing the elements of that science. By Charles Davies, Hartford.
EULER'S ALGEBRA. An Introduction to the elements of Algebra; designed for the use of those who are acquainted only with the first principles of Arithmetic; selected from the Algebra of Euler. Boston.
Peirce's Algebra. An Elementary Treatise on Algebra, to which are added Exponential Equations and Logarithms. By Benjamin Peirce, University Professor of Math. and Nat. Phil. in Harvard University.
Notwithstanding the similarity in the titles of these works, as to the particular end which they severally aim at answering, there is a very considerable difference among them, in respect to the stage of intellectual progress on the part of the pupil, to which they are respectively adapted. In fact this might have been anticipated, from the different positions occupied by the several authors, and the different classes of mind with whose wants they are respectively conversant. Bailey's work seems well adapted to its purpose as an elementary text book for use in schools. It is concise and simple in its directions and explanations, and is arranged in such a manner, and provided with such a selection of examples for practice, as is well adapted to its use as a manual for small classes in the school-room. The analytical method is employed frequently, in detached cases, though the treatise as a whole, cannot be said to be an analytical treatise. It teaches clearly, and succinctly, though in a great degree dogmatically, the more common and fundamental of the algebraic processes. A class of pupils might be carried forward through it, we should suppose, with facility and convenience, as it respects the duties of the instructor, and in the end they would have acquired a considerable degree of familiarity with the elementary processes, and some ideas of the nature of the science as a branch of the great science of reasoning.
The treatises of Davies and Peirce, are somewhat more extensive than Bailey's, and they cover much more ground, so as necessarily to be more concise and compact. They require a higher degree of intellectual advancement in the pupil or more close application in the pursuit of the study. They both purport to be founded in some measure upon the treatise of Boudon-and appear to be highly exact in a scientific point of view. They have still less than Bailey's, any pretension to be considered analytical treatises, in the sense in which that phrase is understood among teachers in this country. The only one in the collection we have before us which can claim this character, is Euler's ; that, however, possesses the character in perfection. It is one of the most simple and beautiful specimens of mathematical analysis in existence. Though composed more than half a century ago, it has still the freshness and beauty necessary to enable it to retain its position for a long time to come. In fact it has very little of the character of mathematical text book ; it is a dissertation on the science, an exposition of principles in a style almost narrative, with all that peculiar charm which genius and science combined, can throw about every subject which they attempt to elucidate. No previous attainments are necessary to read Euler's essay; it commences at the very foundation. But then a certain maturity of mind is necessary in order to appreciate it. Or perhaps it is less a maturity of mind that is required, than a certain readiness ‘at perceiving mathematical relations,-a facility of taking the steps of mathematical reasoning. The study of other treatises gives this readiness, and is therefore a good preparation for the reading of such a work ; but such preparatory studies aid the pupil, not by furnishing him with knowledge w ich is necessary as a preliminary, but only with skill in the power of perceiving mathematical sequences, so as to enable him to follow the author with the readiness and fluency, so to speak, which the peculiar character of the discussion requires in the movements of the mind which it leads along.
Euler was a genius. There is an almost poetical charm in the aspect and relations in which he exhibits even the most abstruse scientific truths. The beautiful generalizations
—the gradual development of truth by the successive steps of an exact analysis, the constant surprises which the reader experiences in arriving unexpectedly upon great and important principles, as he moves along, apparently without any design of seeking them,—these and other characteristics analogous to them, conspire to give to the method of which this treatise is so favorable a specimen, a charm which is almost irresistible.
And yet, although this method has been highly popular in this country, we have never ourselves considered it really adapted to the business of elementary instruction. Uneducated minds cannot move on through such a chain of ratiocination with the readiness and strength necessary to keep their position in mind, and to appreciate the peculiar force and beauty which pertains to such a method of investigation. They must be taught at the outset more dogmatically. The details, the processes, the nomenclature, and even some idea of the very nature of mathematical reasoning, must be acquired by ordinary minds, by slow and successive steps, in which truth is taken first upon trust, and the logical foundations on which it rests are seen afterwards. It might be possible to carry a single pupil, or even a very small class of mature and powerful, and well disciplined minds, over the treatise of Euler, as their first text book ; though we should anticipate the necessity of a 'great deal of explanatory lecturing upon the part of the teacher, and also great difficulty in keeping different minds at all in company ; and after all, the peculiar force and point of the successive steps of the reasoning, all, in fact, which constitutes the peculiar charm and beauty of the method, would probably be very imperfectly appreciated.
A few years ago this method was acquiring great favor, and attempts to extend it to a great variety of branches of instruction were made in all quarters. But we imagine its friends, in forming this favorable opinion, considered more its intrinsic scientific beauty, than its actual adaptedness to the purposes of instruction. There is now an evident tendency to a return to the old mode, in which the various parts out of which the great system is constructed, are taught in detail.-directions are taken upon trust,—the memory is employed to fix them,- practice is resorted to to make them familiar,—and at last the system as a whole is seen and understood at the end, by the combination of elements and parts slowly and somewhat dogmatically communicated. And the great system instead of being presented as a whole, is divided into detached and separate parts, which are brought, one by one, in distinct individuality, before the mind. Expedients and processes are taught separately and expressly,—not brought up incidentally as the difficulties occur which they are intended to remove. In a word, the tendency now is to a return to a method based on its adaptedness to the limited and imperfect and undisciplined minds, whose wants are to be supplied rather than on the intrinsic and absolute nature of the principles of science. There can be no doubt that this return is a judicious one, and we are convinced that a work on the plan of Davies's or Peirce's, will be found altogether better adapted to use as a manual for class instruction than Euler's. We are not sure however that Davies does not go to the opposite extreme. For instance we think a class must be a very extraordinary class indeed to be carried to a state of mind of even tolerable satisfaction and repose, in respect to the whole subject of negative exponents, as it is disposed of on page 16, in a mere corrollary to what will appear to the pupil an accidental example. It is true the proposition a=a", is rigidly demonstrated, but, most pupils, while they might understand, would not feel the force of the demonstration,-for it seems to us there is such a distinction. And then the very perplexing point, as it certainly appears to all beginners, and often to those that are not mere beginners, why such an expression as a-" should be employed as equivalent to an, which forces itself upon the mind at least, seems worthy of some greater atttempt at explanation ; especially when we consider that the doctrine of negative exponents lies at the foundation of so important a part of the subsequent mathematical structure.
We do not mention this as an objection to the book so much as an example illustrating its character. It seems to be condensed, compact, rigid, in the highest degree, It is a text book, and the teacher must supply the commentary.
In respect to the theory of powers and exponents, we have never met with any satisfactory view of it. The different classes of exponents are generally treated entirely distinctly, i. e. so far as the foundation on which the notation is based : whereas they evidently belong to one and the same system, and ought to be brought into the same general