ON LOGARITHMS ILLUSTRATED BY CAREFULLY SELECTED EXAMPLES BY THE REV. W. H. JOHNSTONE, M.A. ASSISTANT-PROFESSOR OF MATHEMATICS AT THE ROYAL INDIAN MILITARY COLLEGE, ADDISCOMBE LONDON LONGMAN, GREEN, LONGMAN, AND ROBERTS 1859 THE recent introduction of Logarithms into the test required for admission to Addiscombe has induced the Author to publish the following treatise. It will be found useful, he hopes, not only to those who are seeking to enter the Royal Indian College, but also to those who intend to continue, or complete, their mathematical studies. The body of the work will be sufficient for students who wish to acquire merely the power of applying Logarithms to arithmetical operations. The Appendices, at the end, are added for those who are desirous of learning the process of constructing Logarithms. For the understanding of the entire work no more previous mathematical knowledge is demanded than that of Arithmetic and the first principles of Algebra. It is perhaps unnecessary to say there is no pretence of novelty in the proofs: but the Author trusts that he has arranged the subject clearly, and methodically, and that he has succeeded in furnishing a serviceable, and a sufficiently copious, set of examples, about the accuracy of which the utmost care has been taken, and to which there are given proper forms according to which the student may work. The advantage, indeed, of cultivating a neat and connected style of writing out logarithmic calculations cannot be too earnestly insisted on. It will be readily admitted by all who have been engaged either in teaching or in examining. The plan adopted in these pages is one that has been recommended by a practical acquaintance with its usefulness. ON LOGARITHMS CHAPTER I. GENERAL PROPERTIES OF LOGARITHMS. 1. IF ax = m, ay = n, a2 = p, &c. where a is a certain fixed number, and m, n, p, &c. are variable quantities, then, The corresponding values of x, y, z, &c. are called the logarithms of m, n, p, &c. respectively to the base a. This may also be expressed thus: where and == loga m, y = logan, z= loga P, &c. m, n, p, &c. are called the natural numbers, It will, of course, necessarily follow that we should say that 1, 2, 3, 4, &c. were the respective logarithms of 3, 9, 27, 81, &c. to the base 3; or we might write, 4 = log, 81, &c. 1 = log, 3, 2 = log, 9, 3 = log, 27, B |