by the tinsel glitter of systems, has ever trusted himself to what Coleridge has so happily termed "the quicksilver mines of metaphysical speculation!" Yet it is important to the best interests of humanity that these mines should be at least occasionally re-opened and worked, and that the dangerous fumes should be encountered by a few for the benefit of all. It is not in human power to discard metaphysics altogether, as Morell, Saisset, and Victor Cousin prove, and as Comte exemplifies by his attempt and failure. The paradox attributed to Aristotle* contains an irrefragable truth: "If we ought to philosophise, we must philosophize; if we ought not to philosophize, we must philosophise: in either case, we cannot help philosophizing." Mr. Morell, following step by step in the footprints of Victor Cousin, has shown that philosophy or metaphysics is a natural and inevitable development of the human mind. As it is concerned with the first principles of our knowledge, it is inextricably implicated with all our reasoning. We cannot divest ourselves of its influence if we would. Let it be recollected that the familiar terms of our ordinary language, substance, essence, being, existence, genus, species, property, difference, accident, general, special, particular, individual, quantity, quality-(we have discarded quiddity and entity, though we retain non-entity)-habit, mode, relation, accident, &c., &c., are strictly logical and metaphysical terms, and that they retain much of their philosophic import, though they have lost their technical precision. From this, it may be judged how impossible it is to exclude metaphysics from even the lower circles of reason and practice. Nay, if our metaphysics be erroneous, the error will ultimately reappear in all our reasonings and actions; and such, we think, is peculiarly the case in the present day. A false and corrupt philosophy has infused a corroding venom into the whole organism of society, and has produced a daily-spreading belief that religion must be rejected as inconsistent with science, while, at the same time, it has fearfully sapped all the foundations of faith. The injury which has been introduced by mistaken metaphysical speculation must be redressed by the juster employment of the same: we must fight the fire with fire; and, following the suggestion of Abraham Tucker, we must cure with the spear of Achilles the frightful ulcer which that spear has occasioned. We doubt, indeed, the possibility of constructing a valid metaphysical system; but a valid metaphysical method, whose re

* Ο Αριστοτέλης ἔν τινι προτρεπτικῷ αὐτοῦ συγγράμματι, ἐν ᾧ προτρέπεται τοὺς νέους φιλοσοφεῖν, λέγει, ὅτι εἴτε φιλοσοφητέον, φιλοσοφητέον, εἴτε μὴ φιλοσοφητέον, piλooopntkov, távtws dè piñoσopnréov.-Prolegg. Phil. David, ap. Aristot. Schol., p. 13, a. 2.

sults might be incorporated with our systems of logic, notwithstanding the heated declamation of M. Saisset, we regard as a desideratum capable of being supplied, and necessary to be supplied, before we can hope for a solution of the mental, moral, religious, political, and social contradictions and heresies which are distracting Europe, and introducing disorder into our own country. We regard the enigmas proposed to our times in these several spheres as imperatively demanding speedy and adequate solution. The relation of philosophy to faith is a question of vital and urgent importance. For this reason we consider that the present age is one in which the renewal of the inquiry into the character and foundations of metaphysics is necessary, and the multitude of philosophers shows that the necessity is recognised. For this reason, too, we have girded ourselves for the task, though unused, and ordinarily disinclined to such speculations. But for our own part, we are willing to adopt the verse of Persius, quoted above, as the motto and colophon of our labours,—our motto as indicating the result of our own and all previous speculation-our colophon, as expressing our own belief that the main requisite of our modern philosophy (which is not deficient in either depth or ingenuity, though it be the deceptive depth and fallacious ingenuity of the ancient Sophists) is to confess its inability to evolve the complete explication of the universe out of the powers of the human mind alone, without the previous confession of its entire dependence upon something higher and indemonstrable, beyond the range of human explanation, whence all the validity of accurate reasoning, and the semblance of truth in all fallacy, are derived. It would be a slander on our own doctrines to pretend that our arguments tend to prove the being of God or the truth of revelation-for these we have declared to be beyond the range of human proof; but the tendency of this whole discussion, we think, is to show the necessity of the recognition of both, not from proof, but from the invalidity of all reasoning, which does not start from their acknowledgment, and the acknowledgment of its own dependence thereon. If this doctrine be once definitely established as a logical pre-requisite of all reasoning, we may then hope to remove the apparent discord between science and religion, which has already proved nearly fatal to the latter-we may harmonize philosophy with Christianity, without imitating Saisset in assigning co-ordinate and co-equal powers to both-we may redeem the age from the charge of a lack of faith, which has been too justly brought against it-we may yet see that reconciliation of reason with faith, which Bayle sighed for, and Leibnitz endeavoured to effect and we may then anticipate, without the arrogant pretension FOURTH SERIES, VOL. III.-14

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of a system of metaphysics, a metaphysical method which may merit to become truly the philosophy of the nineteenth century; and, while glorifying the time by its own glory, may offer some alleviation for the innumerable miseries which have resulted from the ostentatious, sophistical, and blighting pretensions of the selfstyled age of intellect.

ART. II.—THE USE OF MATHEMATICS IN EDUCATION.

"The Logic and Utility of Mathematics, with the best methods of instruction explained and illustrated." By CHARLES DAVIES, L.L. D. Barnes & Co., New-York.

PROFESSOR DAVIES states the object of his work to be "to present the elements of mathematical science separately and in their connexions; to point out and note the mental faculties which it calls into exercise; to show why and how it develops those faculties, and in what respect it gives to the whole mental machinery greater power and certainty of action than can be attained by other studies." In carrying out his plan, he has certainly produced a book of great practical value, if not of the most profound scientific character. Apart from its theoretical views, its practical suggestions, the result of many years' experience as a teacher, will commend the work to all who are engaged in mathematical instruction.

But the question naturally arises, did we need such a work? Is not the world sufficiently satisfied of the importance of mathematical studies? An examination of the course of studies pursued at most of our public institutions, will show that the mathematics have a place in them all; but not by any means the position claimed for them by our author. Indeed, he goes so far as to argue that they are pre-eminently fitted to form the great basis of all education. He asks,

"What system of training and discipline will best develop and steady the intellect of the young; give vigour and expansiveness to thought, and stability to action? What course of study will most enlarge the sphere of investigation; give the greatest freedom to the mind, without licentiousness, and the greatest freedom to action, consistent with the laws of nature and the obligations of the social compact? What system of study is, from its nature, most likely to insure this training, and contribute to such results, and at the same time lay the foundation of all that is truly great in the practical? It has seemed to me, that mathematical science may lay claim to this pre-eminence."

*

Now, we believe that the world thinks more of the immediate practical results of mathematics, than of their importance as a means of training for general usefulness. The mass of men, and even of

educated men, are perhaps too easily satisfied with the results of the labours of the engineer and the architect, and have come to consider these purely physical results as the only desirable ones. The work of Professor Davies will do much to dissipate this idea; and to convince men that mathematical studies most naturally and successfully call into exercise and mature those powers of mind upon which excellence in any and all departments of life depends. He has, perhaps, been less guarded in some of his statements than could have been desired; indeed, the book is calculated, almost, to make the impression that this one line of study would of itself be suffi cient for the adequate development of the human mind in education. But while we cannot go along with this extreme view, we are yet fully of the opinion, that mathematical studies might with great advantage be commenced earlier, and carried further, than they now are in our seminaries and colleges. Without further reference to the work before us, we shall devote the remainder of this article to a brief consideration of the general subject.

The primary object of education being the development and cultivation of the powers of the mind, that course of instruction is least objectionable which tends to call forth naturally, and exercise strongly, all the intellectual faculties; and that student is best educated, who, at the conclusion of his course, has most entire command of these powers. For the attainment of this end it would seem natural that the subjects of study should be varied,—not merely to meet the varied classes of mind to be educated, but to afford expansiveness and play to each individual mind. And there is no just ground for the fear that formerly prevailed, that these separate branches of study cannot be carried on simultaneously without detriment to either. No one in this age of the world is prepared to deny the usefulness of classical training. No other one means of education is to be compared with it, for imparting the power of delicate discrimination in the use of words, or of quick perception of the different shades of thought and feeling expressed by language. But while we would not have these studies neglected, we claim a high rank for mathematics as best adapted to secure habits of attention and accuracy; to develop the powers of abstraction, generalization and analysis; and to cultivate the power of close reasoning, with much that goes to form certainty and strength of judgment.

Locke remarks that, "among all the ideas we have, as there is none suggested to the mind by more ways, so there is none more simple, than that of unity, or one. *** It is the most intimate to our thoughts, as well as the most general idea we have." This idea is the basis of all mathematical science. All the operations of

the different branches of mathematics are but modifications of it. The principles upon which these modifications are based, are so simple and apparent, that they may well be termed axioms, or selfevident truths. They are general propositions; and "are perfectly conceived by a simple process of induction, the moment the facts on which they depend are apprehended." This basis gives force and certainty to all legitimate operations upon the fundamental idea, and is the source of the authority of mathematical demonstration.

In studying the most simple branches of the science, the learner operates upon the most obvious ideas he possesses, by using their simplest relations: to these he can always refer as tests of accuracy; they are few and simple, while the fundamental truths of all other sciences are numerous, complicate, and remote. As the student advances, he finds that every operation implies its own test,-furnishes from its own conditions the means of ascertaining the accuracy of the result. In algebra, the truthfulness of the answer is, to a certain extent, apparent in its very statement. In addition to this, in these operations there are no superfluities; every element has its use and application-every particle, though infinitely small, forms an integral part of the whole, and must appear in the result. Who can fail to perceive how such uses of such principles, must tend to form habits of precision and accuracy?

The power of analysis, as well as of generalization, is developed most advantageously by mathematical training. To separate into elements, to distinguish between parts, and to classify these elements and parts, is the office of the mathematician. The solution of every algebraic equation demands the exercise of the faculty of discrimination, which in other departments of study is so essential to clearness and accuracy. The known and unknown quantities must be distinguished and appropriately arranged; the factors which enter into these quantities must be discussed and classified; and the whole process is, in fact, but the analysis of the truth expressed in the general questions given for solution.

The process of generalization is but the reverse of this operation of analysis, and the power to generalize-to grasp formula which, though brief and concise, shall yet contain all the particular principles on which the operation depends-is justly regarded as one of the most important mental operations. It is useful, not only in every branch of study, but in every department of life. It enables us to express in simple sentences truths, whose elucidation would require volumes. There are no studies which so easily, so naturally, so necessarily lead to the formation of these habits as the mathematics; and the operations of mathematics afford