COURSE OF MATHEMATICS, Theoretical and Practical, FOR THE USE OF SCHOOLS, AND OF THOSE WHO HAVE NOT THE ADVANTAGE OF A MASTER. I. ALGEBRA. II. PLANE TRIGONOMETRY. BY J. R. YOUNG, LATE PROFESSOR OF MATHEMATICS IN BELFAST COLLEGE; AND AUTHOR OF LONDON: G. ROUTLEDGE & CO., FARRINGDON STREET; NEW YORK: 18, BEEKMAN STREET. 1855. 181. C. 128. PREFACE. Ir is intended that the present volume shall be followed by two others, each, like this first, complete in itself, but so connected with the other two, that the whole may form a systematic compendium of modern elementary mathematics. Whether or not this intention be fully carried out, will depend upon the manner in which the portion of the plan now submitted to the public may be received. Should this be such as to encourage the prosecution of the original design to its completion, the three volumes will embrace an amount of mathematical instruction sufficient to meet the demand of most seminaries of education. The volumes in contemplation will contain, in addition to the more advanced parts of pure mathematics, the applications of these to mechanical philosophy, optics, and physical astronomy. The author has not space at his disposal here to give any summary of the particulars introduced into the following pages ; he believes, however, that they may be consulted, not altogether without profit, even by those who are already in some degree familiar with Algebra and Trigonometry. Such persons may, for instance, gather some small additions to their knowledge from the remarks, generally inclosed in brackets, about series-combinations-the incommensurability of the circular circumference, &c. And perhaps one or two things, common to all books of the kind-such as the theory of the greatest common measure, and the investigation of the fundamental formula of Analytical Trigonometry-may be found to be placed in a more agreeable and satisfactory light. But for the reason just assigned, these matters cannot be enlarged upon here and the author can only add that, from the pains taken with the manuscript, and with the proofs, he believes that very few errors indeed have escaped his notice. JUNE, 1855. A COMPENDIOUS COURSE OF MATHEMATICS. I.-ALGEBRA. PRELIMINARY NOTIONS OF THE SUBJECT. 1. THE rules and operations of Algebra are only those of Arithmetic applied to letters as well as to figures. You will gain a notion of how letters can be introduced into calculation as well as figures, by reflecting upon what is done in working a common addition or subtraction example in the arithmetic of compound (or concrete) quantities; as for instance, in adding together several sums of money, or in subtracting one sum of money from another. Thus, the operation in the margin, though one of common arithmetic, differs but very little from addition of algebra; and, as far as the part performed by the letters is concerned, it does not differ at all. ADDITION. 261 38 2d 177 1s 3d 431 28 4d 91 58 ld 78 951 188 10d The letters 7, 8, d, in this example, are symbols, standing for pounds, shillings, and pence. It is universally agreed that, in money operations, these symbols shall have the meanings here stated, and no other meanings; yet it is plain that if any person chose to do so, he might employ these same symbols l, s, d, in his own private practice, to stand for days, hours, and minutes; or indeed for any other quantities. The result of the operation above-taking the symbols in the commonly received sense is 95 pounds 18 shillings and 10 pence; but had these symbols stood for days, hours, and minutes, as just supposed, then the result would have been interpreted 95 days 18 hours and 10 minutes. In working the above example, as in common arithmetic, the symbols 1, s, d would not be written against each of the several items as here; they would of course be placed over them once for all: they have been repeated in this place solely for the purpose of giving an algebraical form to the example, and having gone over it, you may consider that you have worked an exercise in addition of algebra. It is true that if this exercise were given among algebraical examples, the B |
