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templation by the operations of nature, made it necessary to illustrate our course very fully by problems entirely of a hypothetical kind : thus, as examples of the theory of central forces, our books were filled with investigations respecting laws of force, which, to say the least, are not known to exist in nature ; and although the conclusions deduced might be perfectly satisfactory on the assumed hypothesis, they could not be considered of the slightest use in a philosophical point of view. They were mere exercises, by which it was intended to ensure the student's right understanding of the propositions they were meant to exemplify.
In this way they may still be valuable, provided they follow sufficiently directly from the general theory to enable the student himself to deduce them from it; but if they are such as to require his particular and undivided attention in themselves, or are so far removed from the general course of his studies, that he must obtain his competency in these particular cases by careful perusal of the works of others, they lose their value as examples or illustrations, as far as he is concerned; and therefore, if they are unconnected with the operations of nature, the time devoted to them must be considered thrown away.
It may be urged that they have a use simply as exercises of his reasoning faculties; but, in the present state of science, so many problems and illustrations may be presented to him, which, besides this use, will give him considerable insight
into the connexion of mathematical science with the operations of nature, that surely the latter class should always be preferred.
Before the problem of the three bodies was understood with any degree of accuracy, we wanted illustrations of the theory of central forces. The problems depending on this subject presented to us by nature, were of too complicated and abstruse a kind to act as such, our course of pure analysis being quite incompetent for their solution.
To supply this deficiency the investigations of the orbits commonly called Cotes' Spirals, were introduced, and for a long time were treated as a necessary part of our academic course; but since the Lunar and Planetary theories have been more fully developed, while our knowledge of pure analysis has been so much extended as to enable us to make very considerable progress in these natural illustrations of the theory of central forces, Cotes' Spirals have been properly considered in the light of mere mathematical curiosities, and as such, injudicious applications of the student's valuable time.
In like manner, now that hydrostatical science enables us to gain considerable insight into the theory of sound, the student should never be encouraged to apply his time to learning to find the different positions in which a triangle, if kept edgewise in a fluid, would float in it; nor in general would it be advantageous that those exertions which might be devoted fruitfully and agreeably to the theory of light, achromatism, and all their
beautiful phenomena should be lost in finding caustics to reflecting and refracting surfaces, which as yet we are not able to make experiments with.
It is gratifying to see, by inspection of the following examination questions, that the views above stated have been acted upon to a considerable extent; and perhaps they would have even more influence on our course of reading, were they to be more formally and generally recognised. The main thread of our investigations would not (as is sometimes the case in books published for students in the University) be continually interrupted by problems of too difficult a nature to act as illustrations, and serving rather to puzzle than enlighten the reader.
If we can make sure that the student clearly apprehends the nature of the general processes and essential theorems of a science, by frequently presenting to him sufficiently full, yet easy examples of them, we have gained a great step, and he may safely be trusted to pursue those higher branches which are more directly connected with the philosophy of nature; while, by constantly calling upon him to get up problems of a useless yet extremely difficult kind in the lower branches, we run the risk of encumbering his memory without invigorating his mind, and stopping him short in his course long before he has caught a glimpse of the real beauties of mathematical science, or seen one instance in which it is made available to unravel the secret operations of nature, and display the beautiful contrivances by which the universe is adapted to our wants.
It must be admitted, that to arrive at the degree of mathematical proficiency requisite for the investigations of physical astronomy and other 'natural applications, will under the most favorable circumstances require considerable industry and mental exertion; and it is from this fact, coupled with the too frequent misapplication of the student's time, that so many persons are to be found who decry the study of mathematics as uninteresting and practically useless.
The course of pure mathematics, must, necessarily, be extensive and frequently abstruse; the extreme complication of the causes by which the phenomena of the universe are effected, demands of us an industrious, and as far as possible, complete investigation of all those propositions which concern magnitudes generally. Nature frequently presents to us problems, which would require for their solution, an analysis, even more powerful than any yet known; whence it is justly inferred, that no investigations which tend to enlarge our knowledge in pure analysis can be deemed, a priori, void of utility; but even here it may be advantageous in a course intended to instruct and
open the mind, to include at first only such parts as can be fairly applied to the researches of natural philosophy.
With respect to what are termed the mixed mathematics, the case is far different. It may be boldly asserted of many of the difficult hypothetical problems contained in some elementary works, not only that they are at present inapplicable to
natural cases, but that they must always remain so; and surely these can never be considered part of the essentials of science, and consequently should never form subjects for questions in our book-work papers.
The Editor trusts, that a perusal of the present compilation will shew, that these views are not peculiar to himself, and that the conviction he entertains of their beneficial influence upon the course of study pursued at the University, and of the advantage that would result from their being still further acted upon, will be deemed a sufficient excuse for his offering the above observations on the subject.