THIS book is intended as a course of Elementary Pure Mathematics, for the special use of those preparing for the Government Science and Art Examinations. It is not intended for very young boys, inasmuch as a certain amount of preliminary knowledge is expected from the student; and the explanations and examples might be held to be too difficult for those who had not at least laid a foundation of arithmetical science. But for such as usually submit themselves to the test of the "elementary stage," it is hoped that this volume will prove a serviceable and sufficient text-book. The compiler has endeavoured to enunciate sound principles, neither more nor less than are necessary for the purpose, and to illustrate these principles in the clearest manner. In particular, he has tried to render the work acceptable as a teacher's handbook, being convinced that oral class teaching is infinitely more valuable, as it is certainly becoming more and more customary, than a painful process of book-poring on the part of the student. L. S. LONDON, December, 1872, MATHEMATICS. ARITHMETIC. Numeration and Notation. 6 1. If we set down one figure, or several figures side by side, we call that which we have set down a number. Thus 3,472,569 is a number, and represents 'three million, four hundred and seventy-two thousand, five hundred and sixty-nine.' The process of representing the number by words is called Numeration. The process of representing the number by figures is called Notation. 2. We commonly employ nine figures (in addition to O for 'nought' or zero'), because in our decimal scale of notation we have other means of representing ten and all higher numbers. The number above given will serve as an example. The 9, in the right hand place, represents nine units, or ones; the 6, in the second place from the right, represents 6 tens; the 5, five hundreds; the 2, two thousands; the 7, seven tens of thousands; the 4, four hundreds of thousands; the 3, in the seventh place from the right, three millions; and so on, until a figure in the thirteenth place would represent so many millions of millions, or billions. If, now, to the number above given we add 1, or one unit, we shall have altogether ten units, that is, one ten. This ten must go in the tens' place; and the number would end with 570, 'five hundred and seventy,' instead of 569. Similarly, if we had added to the same number one million, five thousand, two hundred and one—that is, 1,005, 201, we must write the result 4,477,770-that is, 'four million, |