The work is not a mere compilation. The subjects treated of are necessarily much the same with those which may be found in other writers; but the arrangement, the language, and the examples, are wholly our author's. And if clearness of method, a judicious selection of materials, and perspicuity and neatness of expression, be regarded as furnishing any claims to originality, we think Professor Day is, in no common degree, entitled to the character of an original author.- The general principles, which are printed in a distinct character, and are designed to be committed, possess a degree of brevity, clearness, and precision, which we have seen in no other mathematical work. His illustrations are generally somewhat diffuse; but uniformly luminous, and adapted to the capacity of the learner. He appears to us to possess, in a degree unusual in men of profound science, the power of placing himself in the attitude of a learner, of feeling his difficulties, and of hitting on the most happy expedients for removing them. When an obstacle is to be removed, the most advantageous position is assumed, and the lever uniformly applied at the right spot. As examples of this fortunate mode of illustration, we might refer our readers, among many others, to the distinction given between positive and negative quantities,-to the explanation of the reduction of a problem to the language of Algebra, to the remarks on the reduction of affected quadratics,-to the illustration of mathematical infinity,-or, in the last Number, to the view given of the principles of Mercator's chart. Here, if the adept in science finds nothing absolutely new, he will at least find known truths placed in a stronger light, and happier attitude. In regard to his illustrations, our author has taken for his model the familiar, diffuse manner of Euler and Lacroix, rather than the concise, abridged mode of the English writers. The reasons which have induced him to carry his explanations to a greater length than most elementary writers have done, may be found in the Preface to the Algebra. Although they appear, on the whole, sufficient, this method is liable to an objection which our author has not noticed. He has, indeed, furnished us with a highway up the difficult ascents of mathematical science; in travelling which the student of ordinary talents and diligence will find little to impede his progress: But some may question whether an obstacle occasionally left to be removed by his own exertions,a step in the ascent required to be dug by his own labour,will not ultimately contribute to accelerate his march. Admitting that hours may be spent, in supplying an explanation, or an article of proof, which, if it had been inserted in its place, might have been read and understood in a few minutes,' we can by no means consider this time as absolutely wasted. The student has been concentrating all his powers on a single point. He has been obliged to summon to his aid all his past acquisitions; from which his sagacity is exercised, in selecting and applying the proper media of proof. The satisfaction of finding one difficulty surmounted by his own exertions, will inspire him with new vigour, and confidence in his ability to overcome others. The illustration which is read and understood in a few minutes, may be almost as soon forgotten: But those conclusions which are the result of hours of active labour, on the part of the student, will never be forgotten. At the same time, the intermediate truths which he was obliged to call to his aid, will be associated in the memory with the final result. The utility, as well as the interest, of scientific studies, depends much on the degree in which the mind is rendered active, in pursuing them. We do not hesitate to say, that what is generally termed a taste for particular branches of study, is rather the gradual result of the pleasure attending the successful exercise of our faculties in those pursuits, than of any diversity in the original constitution of different minds. We would not, with our author, object to abridged compilations, therefore, on the ground that the supply of their deficiencies will cost the student additional time and labour; but because few can, in fact, be prevailed upon to submit to this labour. Could the great body of students at college be induced to supply for themselves the chasms in such systems as that of Webber, we conceive that their time could not be better employed; and that, although their first advances might be slow, their final progress would be more rapid than by any other method of study whatever. But, at the age when such a course is usually put into their hands, few pos sess sufficient perseverance to remove the difficulties which lie at the threshold; and, instead of forming the habit of inventing demonstrations for themselves, and pursuing untrodden paths with pleasure, they contract a disgust for the whole subject. * * What Hederic says, respecting the Analytical Part of his Lexicon, may, mutatis mutandis, be applied to the subject under consideration. Altera illa, (Pars), cui Analyticæ nomen feci, ea Vocabula grammatice, quomodo loqui consuevimus, resoluta complexus sum, quæ difficultatibus suis, sive a Dialecto aliqua, sive aliunde subortis, solicitare tironum patientiam possunt.--Ac temetsi non desint, qui eamdem ignaviæ potius sub No one will infer, from the foregoing remarks, that we consider the mind as entirely passive, in pursuing such full illus. trations as those of Mr. Day, or the complete demonstrations pure Geometry. Close attention, and considerable exertion, are doubtless requisite, in tracing the investigations even of analytical and synthetical demonstrations, which are so full as to leave nothing to be supplied by the ingenuity of the student. All we contend for is, that invention is a more important exercise, and is attended with more rapid improvement. We are inclined, on the whole, to give the preference to the full method of illustration which Mr. Day has adopted. It will not produce as many original, inventive mathematicians as an abridged mode, which leaves more to be supplied by the student; but it will furnish many more with some share of that intellectual improvement which is the object of the mathematical part of a collegiate course. And the author, with propriety, adopts a more concise method, as he advances in his work. Paradoxical as it may appear, we are not confident that Mr. Day has not carried his principles of rendering mathematics easy, somewhat too far. If the symmetry of the fabric would be impaired, by any essential omissions, either in the demonstrations or in the analytical parts, might not some of those incidental propositions which form no part of the body of the work, be intentionally left with advantage, to be demonstrated by the learner? The miscellaneous problems occasionally inserted might have been made more numerous and difficult. If, in addition to these alterations, the particular applications of general principles, which have been demonstrated, were, in more instances, left to be made by the learner,-if, for example, after demonstrating the general principles of rectangular trigonometry, and exemplifying them in a sufficient number of cases, to render the application familiar, the rest had been merely enumerated and the proportions required of the student, —although more time might be occupied, no loss would probably, on the whole, be sustained. sidium habeant, quam ut emolumenti quidquam juventuti illam credant adferre, eo quod hæc a penitiori Grammatices studio evocet, ad quod potissimum solidior linguæ cognitio redit: attamen, qui rectius justiusque rem expendunt, satius utique existimant, velificare nonnihil discentium a labore molestiore aversioni, quam quidem commitere, ut miseri succumbant insitiæ suæ, vel etiam, quod sæpius usu in Scholis evenire videmus, difficultatibus his abterriti linguæ pulcherrimæ studium plane abjiciant.'Præf. Græc. Lex. Our author's selection of materials is, in general, judicious. Little will be found in the text, which is not strictly elementary, and practically important,-at least from its relation to subsequent parts of the course. At the same time, at least as much matter is comprised under the several heads, as can be attended to in the time usually allotted to mathematical studies, in our public seminaries; much more, we are persuaded, than has hitherto been usually studied. In addition to this, the notes, at the end of each Number, contain a mass of valuable matter, collateral with, and explanatory of the subjects contained in the text; and they will be read with interest by those whose curiosity leads them beyond the limits, to which the text is confined by its original design. Throughout the work, an important principle is, with very few exceptions, observed. This is, to demonstrate every truth when it is first asserted, and never to take for granted principles, the proof of which depends on subsequent articles. This circumstance, together with the constant system of reference, which is kept up, to preceding articles, gives this work the aspect of a regular body of science, in a higher degree than any other with which we are acquainted. It is no inconsiderable advantage, too, that the several steps of each process are placed under each other, and numbered. Instead of presenting a congeries of algebraic characters dispersed over the page, which is too much the character of many analytical works, each successive step is distinctly presented to the eye, occupying, with a brief explanation on the left, a line by itself. By the aid of the numbers, any pre ceding step in the process, whatever be its length, may be readily referred to. The terseness and symmetry with which the matter is arranged on the page, we cannot too much admire; but we regret to see so many contractions as are sometimes made, to bring several related particulars to the same form: and we had rather see some sacrifice in regard to the symmetry which these particulars exhibit, than the deformity and obscurity, which such contractions sometimes occasion After this view of the general character of Professor Day's Mathematics, our readers may expect a more particular examination of its contents in succession. The first thing which will probably occur to them as needing explanation, is, that, while it professes to be a Course of Mathematics, adapted to the method of instruction in the American colleges, it contains nothing on the elements of Arithmetic. No reasons are assigned by our author for this omission; and it can be justified only on the ground, that Arithmetic will be required as a condition of admittance into our public seminaries. An accurate acquaintance with this science undoubtedly ought, if possible, to be made a requisite for such admission; but we do not believe that it will soon be practicable, to such an extent as to supersede the necessity of afterwards revising it. In those grammar schools, where the preparatory classical studies are made the primary object of instruction, Arithmetic will be thrown into the back ground, and will now be taught in that thorough, practical manner, without which the student must be utterly unfit to enter upon the higher branches. In the present state of education, it appears to be indispensable, that every class of students in college should revise the principles of Arithmetic, and be familiarized to arithmetical operations, in concert. Indeed, we can hardly conjecture what was our author's motive in excluding Arithmetic from his system, unless it was a supposed difficulty of demonstrating its principles, without violating a rule which he had prescribed to himself, by anticipating the more general symbols and reasonings of Algebra. But why cannot Arithmetic have its peculiar mode of demonstration, as well as Algebra and Geometry? The simple and local values of figures ought to be made subjects of definition; the proper axioms ought to be laid down; and the truth of the several operations with numbers would be as susceptible of demonstration, as that of the common solutions of geometrical problems. As particular diagrams are used in Geometry, to represent all magnitudes of the same class, so, in most cases, particular numbers may be used in Arithmetic, to demonstrate what is equally true of numbers in general. The common method of demonstrating arithmetical rules, we are sensible, has been, to refer them to the more general principles of Algebra: and, to those who are acquainted with Algebra, this is doubtless the best mode of exhibiting the truth of the series of simple operations, which constitute a rule in Arithmetic. But this method of demonstration cannot be extended to the simple operations themselves. As is observed by Legrange, one of the greatest analysts of modern times, the province of Algebra is not actually to find the values of the required quantities, but to trace the system of operations by which they may be deduced from those which are given.'* This • system of operations contained in the final formula, if ex. pressed in common language, becomes an arithmetical rule. But the application of this rule, to the finding of a particular * Theorie des Equations. |