truth of our nature, and if he does not find faith on the Earth, he brings it. However delayed, the word of LIFE is sterling; and the Lord of the Sabbath day, having set doing good to man against every institution. made for man, shall come again in glory; and as the lightning cometh out of the East, and shineth even unto the West, so shall that coming of the Son of Man be! E. P. P. ART. IV. - An Elementary Treatise on Algebra, for the use of Students in High Schools and Colleges. By THOMAS SHERWIN, A. M., Principal of the English High School, Boston. Boston Benjamin B. Mussey. 1842. "To the making of books there is no end." We are glad of it. Why should there be an end? If a book tend to facilitate the acquisition of knowledge, either by presenting new information, or by serving up old in a more attractive form, it ought to be welcomed; and it will be by those, whose pecuniary or other exclusively selfish interests are not liable to be injured by its introduction. We have no objection to a book, because there are already a score of a similar kind in the market. We have no objection to a book of instruction, even if it is made with a pair of scissors, provided the instruments are judiciously employed, to cut out such well treated portions of the subject matter, as best announce and illustrate it. We seldom find a single author treating all the branches of a subject equally well; and it therefore becomes important, especially to the young, that some discriminating person occasionally collect, into one compact treatise, the isolated improvements, which have been made by successive authors, and which have stood the test of experimental instruction, during a course of years. The more these portions are fashioned into symmetry, and receive new life and spirit from the compiler, the better will his book fulfil its design. We do not mean to insinuate, that the book before us is a mere compilation; for, although in it, as an elementary treatise, we find only long established principles, yet the exhibition and demonstration of them, as well as their arrangement and the mode of instruction, (considerations of the utmost importance to the learner,) are original. The object of this notice is, to state what we consider the requisites of a good elementary instruction book of Algebra, a science, dignified in itself, and so important in its various applications, that it surely makes legitimate claims to well digested treatises and excellent manuals for instruction. Mathematical knowledge seems to be gaining a more extended diffusion in our country; and we hope, at the same time, that the standard of mathematical acquirements is steadily and rapidly rising. Our common schools, which were formerly considered in good condition, when a few of the most advanced pupils, (by dint of sundry mystical operations prescribed, not explained,) had succeeded in "getting through," Daboll's arithmetic, now send forth their annual crowd of boys. and girls, at an age much younger than of yore, not only better acquainted with arithmetic, but often with a very useful, though doubtless a limited knowledge of algebra and geometry. It seems fair to infer, that, the common schools being thus indisputably better than formerly, the higher schools and colleges have risen also; but the fact, in regard to those high schools in which boys are prepared for college, is, that Latin and Greek are the principal objects of attention, while barely enough of arithmetic and algebra is acquired, to pass an examination at college; consequently, the boys, not being thoroughly prepared in elementary knowledge, find themselves unable to accomplish the course of mathe matical studies prescribed by the college faculty. These facts show, that at least one description of high schools has not been very successful in raising the standard of mathematical knowledge; and, what is worse, has consequently impeded its advancement in the colleges. The blame, we suppose, must be shared between teachers and parents; the teachers for not insisting on keeping back the unprepared pupil; and the parents for excessive eagerness to get their sons into college. Hurry, however, is our national characteristic. The text books of our colleges have, within a few years, been changed for the better; and doubtless individual students are occasionally found, who have made progress far beyond those of the same rank and pretensions in former times; but, the "rari nantes in gurgite vasto" are few indeed, compared to those, who sink down through utter weariness and disgust; for we believe it to be an indisputable fact, that there prevails in our colleges a very general and lamentable disrelish for mathematics. There are other causes of this disrelish, besides the insufficiency of preparation; such are, the great amount of time and labor requisite for proficiency, the supposed improbability of making an every-day use of them in the professions, the small chance they offer to one's vanity as a means of display. As to a want of natural capacity, we cannot believe this to be quite so general as the dislike, and, therefore, cannot set it down as peculiarly influencing this branch of knowledge. The most common complaints, which the students make openly, are, "the books are too difficult; we are driven on too fast; " &c. Now many utter these complaints, who are glad to find any plausible pretence to cover their real indolence and negligence; and yet, we believe, the rogues have sagacity enough, in the present instance, to use a just discrimination in their choice of pretences. We do truly believe, that many of the mathematical text books, in our colleges, are too difficult for a great majority of the persons called upon to use them. The treatises of Professor Peirce, for instance, elegant and symmetrical as they are, are too condensed and general for boys; they presume a greater preparation in mathematics, than boys usually get before going to college, and greater than the examinations for admission seem to demand; in fact, they appear to have been made, with a prospective view to a much higher standard of mathematical acquirements, than the present state of things in our schools and colleges immediately promises; and, we believe, they are thoroughly perused and understood but by a very few collegians, and these remarkably well endowed with faculties for mathematical studies. It may be said, that the subject demands this great generality; that it is its chief virtue; is all, perhaps, that renders it valuable in the higher departments of science dependent on analysis; and if boys do not readily show talents for such reasonings, they must leave the subject to heads better constituted by nature for mathematical investigations. In reply to this, we declare, that children must be fed "with food convenient for them," with children's food, so diluted as not to clog by repletion, and yet so strong as sufficiently to stimulate their faculties to healthy activity. Boys must be trained in a long course of exercises in the elements of the exact sciences, must be made perfectly familiar with their instruments, before they can be justly expected to grapple with general views and discussions. Many people, and especially persons of great natural powers for acquiring mathematical knowledge, forget how slowly young persons attain an acquaintance even with arithmetic; that not one in twenty, of the boys admitted into our colleges at the age of fifteen, nay, of the whole that enter, inclusive of those whose age approximates thirty, has a thorough and firmly settled knowledge of this first branch of mathematics, although they have, on an average, attended to the subject ten years. It would appear at first glance, that, if the very elements of science can scarcely find a lodgment in their crania, after so long, so Troy-like a siege, it were ad visable to desist; but many a dull boy, by patient perseverance and accurate instruction, has finally arrived at an enviable acquaintance with the deepest science, quæ, sera, tamen respexit inertem, Candidior postquam tondenti barba cadebat. We do not wholly approve of setting up a procrustean system, and declaring that books should be made only in accordance with it; but, in an instruction book of an exact science, we can tell, a priori, what are indispensable requisites; and we claim a right to examine such books, with reference to our preconceived notions of what they ought to be. We proceed to mention some. ax + by xy The first requisite, in the formation of an instruction book, is, that the author, being thoroughly acquainted with his subject, invariably tell the truth. One of the most popular elementary instruction books of algebra, and that made by a teacher of great celebrity and success, contains in its key, as explanatory of the process by which the fraction may be reduced to its lowest terms, the following morceau; "since ar occurs in the numerator and denominator, this term may be cancelled in both. Ans." The enunciation is too formal and express to allow of the charitable construction of a lapsus, and the subsequent reduction of similar quantities, by the same rule, settles the blame of ignorance of a first principle on the author. How he could ever have made progress in analysis, with such notions of reduction, is a puzzling quest ion. a + c From this single instance of false teaching, all will readily see the absolute necessity of stating truth alone, in elementary books; as an error, imbibed early and habitually assumed, will constantly recur and annoy the student, even after a better acquaintance with the subject has exposed its falsity. The truth should also be so strictly observed, that no statements, made in general terms, for the sake of producing a strong impression on |