IN compiling the following pages, my object has been to notice particularly several points in the principles of Algebra and Geometry, which have not obtained their due importance in our elementary works on these sciences. There are two classes of men who might be benefited by a work of this kind, viz., teachers of the elements, who have hitherto confined their pupils to the working of rules, without demonstration, and students, who, having acquired some knowledge under this system, find their further progress checked by the insufficiency of their previous methods and attainments. To such it must be an irksome task to recommence their studies entirely; I have therefore placed before them, by itself, the part which has been omitted in their mathematical education, presuming throughout in my reader, such a knowledge of the rules of algebra, and the Theorems of Euclid, as is usually obtained in schools. It is needless to say that those who have the advantage of University education, will not find more in this Treatise than a little thought would enable them to collect from the best works now in use, both at Cambridge and Oxford. Nor do I pretend to settle the many disputed points, on which I have necessarily been obliged to treat. The perusal of the opinions of an individual, offered simply as such, may excite many to become enquirers, who would otherwise have been workers of rules and followers of dogmas. They may not ultimately coincide in the views promulgated by the work which first drew their attention, but the benefit which they will derive from it is not the less on that account. I am not, however, responsible for the contents of this treatise, further than for the manner in which they are presented, as most of the opinions here maintained have been found in the writings of eminent mathematicians. It has been my endeavour to avoid entering into the purely metaphysical part of the difficulties of algebra. The student is, in my opinion, little the better for such discussions, though he may derive such conviction of the truth of results by deduction from particular cases, as no à priori reasoning can give to a beginner. In treating, therefore, on the negative sign, on impossible quantities, and on fractions of the form &c., I have followed the method adopted by several of the most esteemed continental writers, of referring the explanation to some particular problem, and shewing how to gain the same from any other. Those who admit such expressions as ·a,√ - a &c., have never produced any clearer method; while those who call them absurdities, and would reject the altogether, must, I think, be forced to admit the fact, that in algebra the different species of contradictions in problems are attended with distinct absurdi ties, resulting from them as necessarily as different numerical results from different numerical data. This being granted, the whole of the ninth chapter of this work may be considered as an inquiry into the nature of the different misconceptions, which give rise to the various expressions above alluded to. To this view of the question I have leaned, finding no other so satisfactory to my own mind. The number of mathematical students, increased as it has been of late years, would be much augmented if those who hold the highest rank in science would condescend to give more effective assistance in clearing the elements of the difficulties which they present. If any one claiming that title should think my attempt obscure or erroneous, he must share the blame with me, since it is through his neglect that I have been enabled to avail myself of an opportunity to perform a task which I would gladly have seen confided to more skilful hands. ON THE STUDY AND DIFFICULTIES OF MATHEMATICS. CHAPTER I. Introductory Remarks on the Nature and Objects of Mathematics. THE object of this Treatise is-1. To point out to the student of Mathematics, who has not the advantage of a tutor, the course of study which it is most advisable that he should follow, the extent to which he should pursue one part of the science before he commences another, and to direct him as to the sort of applications which he should make. 2. To treat fully of the various points which involve difficulties and which are apt to be misunderstood by beginners, and to describe at length the nature without going into the routine of the operations which have been already discussed in the Treatises of Arithmetic, Algebra, and Geometry, published by this Society. માઇ thing in the world that the account, or any material part of it, should be false. This is perfectly correct, nor can there be the slightest objection to believing the whole narration upon such grounds; nay, our minds are so constituted, that, upon our knowledge of these arguments, we cannot help believing, in spite of ourselves. But this brings us to the point to which we wish to come; we believe that Cæsar was assassinated by Brutus and his friends, not because there is any absurdity in supposing the contrary, since every one must allow that there is just a possibility that the event never happened: not because we can show that it must necessarily have been that, at a particular day, at a particular place, a successful adventurer must have been murdered in the manner described, but because our evidence of the fact is such, that, if we apply the notions of evidence which every-day experience justifies us in entertaining, we feel that the improbability of the contrary compels us to take refuge in the belief of the fact; and, if we allow that there is still a possibility of its falsehood, it is because this supposition does not involve absolute absurdity, but only extreme improbability. No person commences the study of mathematics without soon discovering that it is of a very different nature from those to which he has been accustomed. The pursuits to which the mind is usually directed before entering on the sciences of algebra or geometry, are such as languages and history, &c. Of these, neither appears to have any affinity with mathematics; yet, in order to see the difference which exists between these studies, for instance, history and geometry, it will be useful to ask how we come by knowledge in each: suppose, for example, we feel certain of a fact related in history, such as the murder of Cæsar, whence did we derive the certainty? how came we to feel sure of the general truth of the circumstances of the narrative? The ready answer to this question will be, that we have not absolute certainty upon this point; but that we have the relation of historians, men of credit, who lived and published their accounts in the very time of which they write; that succeeding ages have received those accounts as true, and that succeeding historians have backed them with a mass of circumstantial evidence which makes it the most improbable In mathematics the case is wholly different. It is true that the facts asserted in these sciences are of a nature totally distinct from those of history; so much so, that a comparison of the evidence of the two may almost excite a smile. But if it be remembered that acute reasoners, in every branch of learning, have acknowledged the use, we might almost say the necessity, of a mathematical education, it must be admitted that the points of connexion between these pursuits and others are worth attending to. They are the more so, because there is a mistake into which several have fallen, and have deceived others, and perhaps themselves, by clothing some false reasoning in what they called a mathematical dress, imagining that, by the application of mathematical symbols to their subject, they secured mathematical argument. This B |