Cash Merchant Tailoring and Clothing Empori No. 237 NORTHAMPTON STREET, EASTON, PENN’A, Between Second and Third Streets, Has constantly on hand the largest assortment of PIECE GOODS in Easton. The most fastidious can be suited. All work guaranteed to please the wearer. Every variety of Toilet Articles, Perfumery, Soap, Brushes, &c., &c. Especial attention given to compounding Physicians' Prescrip- REFRESHMENT PARKOR. OYSTERS DELIVERED TO FAMILIES FREE OF CHARGE. THE LAFAYETTE MONTHLY. Editors for January.-W. H. BAYLESS, J. B. HELLER, JR., C. J. NOURSE. VOL. IV. JANUARY, 1874. NUMBER 5. COLLEGE MATHEMATICS. BY R. J. WRIGHT. I. In general. In considering the study of mathematics for the purposes of a college course, and especially for those modern and liberal purposes, wherein different courses are allowed, we have to consider the objects aimed at in the use of the study, These objects divide themselves into two. One object is to use the study as a discipline of the mind for other studies. The other object is to acquire an amount of mathematical knowledge for ultimate mathematical uses, as for engineering, astronomy, mathematical professorships, and so forth. The usual college course of mathematics has been established with a view to both these ultimate uses, and therefore has not been differentiated (Spencer's use of the word differentiated) for either of them; just as chemistry and pharmacy were formerly taken together. Even the polytechnic schools have made no scientific differentiation in reference to the two ultimate uses, and for their own particular use. Those schools have merely dropped out of the course, the languages and the moral and metaphsical sciences. So also the old colleges and universities, which furnish special scientific courses, have not yet made a scientific differentiation between the two uses. They have only changed from the ancient languages to the modern languages, and pushed the mixed mathematics harder in the latter years of their course. And yet the scientific and technical courses in the old universities are only a little more than the primitive idea of elective fully developed. Now what is needed is a radical reform in the matter, a going back to first principles, and differentiating the two courses of mathematics entirely anew with a view of their ultimate uses, namely of discipline or of direct professional application. II. As to the use of mathematics for discipline, little need be said in its praise to those who have tried it. For those who have not tried it, and have not faith in those who have, the all important means are to simplify mathematics so they can understand it, and at the same time to pick out its most interesting ideas and processes, so as to attract them. And happily these two means are entirely at one, and consist in picking out the general principles and omitting details. Because in all the sciences the general principles are the easiest and the most interesting; and from the fact of being the most general are of course the most disciplinary to the mind. Geometry should come earlier in the course than algebra; because it is better as a logical discipline, and because it is less abstract, as it always has diagrams before it and has reference to them, and also because its beauties and its successive uses can be more easily made apparent. The next means of improving the disciplinary use of mathematics has reference to its use as a training for logical argument. For this use geometry is almost everything, algebra almost nothing. Hence a thoroughly differentiated course for the use of discipline would omit algebra entirely, except a few of its general ideas easily communicable in half a dozen lessons, and then a few lessons of simple examples for the sake of its original analytic training. Other analytical training being sufficiently obtained in the analytical geometry. Now as to the method of studying geometry for a discipline, there comes the necessity for the greatest of changes. Every demonstration should be a "speech" and an oration, both in its rhetoric and its declamation; and to this rhetoric and declamation quite as much attention and criticism should be devoted as to the other orations and "compositions" in the course. Our next thought is that this disciplinary use of mathematics requires a little study of the general principles of every branch of the pure mathematics; the disciplinary points being to give the student a thoroughly clear idea of what each branch thereof is. The discipline here first consists in the clearness and generalness of the idea of each of the parts separately. Next there is wanted a thoroughly clear general idea of mathematics as a whole, which idea can only be obtained after obtaining the true idea of the parts separately. But no disciplinary use of any study can be complete, only in proportion as it drops out of its steps and processes, everything that is merely temporary or merely scaffolding, and intended to be afterwards forgotten. A training to forget, or which is the same thing, the study of anything which is expected to be afterwards forgotten, is corruptive of true mental discipline. This is another reason why thoroughness, without regard to quantity, should alone be aimed at in this use. Our last idea of mathematics as a discipline, and especially as a discipline for argument, is that its nature must be investigated, its functions in comparison with, as well as interwowen in all the sciences must be shown and explained. In other words not only the rationale, but the reasons of mathematics must be given. All this means that the philosophy of mathematics is the culminating point of its disciplinary use. This, of course, embraces a new but brief travel over the whole science from the numeration table up; because numeration and fractions, being thoroughly understood, is quite half preparation for the calculus. Finally as to discipline; discipline means that studies aid, and that a scholarly course of studies are coming after it. And a gentleman and scholar will be expected to know something of mathematics. This expectation requires a knowledge of trigonometry with brief application of the general principles to surveying, navigation and astronomy. All the work of a few lessons. III. The two courses being then differentiated, we have now to consider the course intended for direct mathematical uses. Observe then that this course being rather excessively mathematical, cannot be pursued successfully without bringing a high degree of mathematical, training, apt indeed to give the mind a one-sided bent. Hence in this course the object of discipline therein may be overlooked. Nay more the mathematical discipline ought rather be reduced to a minimum, consistent with the large amount of mathematical knowledge to be acquired. Hence then the headings of the propositions in geomety may be committed to memory, and their meaning made familiar, instead of the demonstrations being studied; except perhaps the simpler and |