The considerable additions and alterations made in this Edition do not consist in the introduction of any new methods, so much as in the further development of those already laid down in the first Edition. The compendious method of Division is now used throughout the book, and is introduced in the process of finding the Greatest Common Measure; the contracted method of Multiplication and Division is likewise extended to compound quantities. The growing importance of the question of a decimal and an international coinage, and of a more uniform system of weights and measures has led to the introduction of a new Chapter on that subject: while the Chapters on duodecimals and on the extraction of the square and cube root have been recast. A very large addition has also been made in the Appendix to the collection of Examination Papers; and for permission to publish these I am indebted to the courtesy of the Masters of the several Colleges and Schools wherein the Papers were originally set. ARITHMETIC. CHAPTER I. FIRST PRINCIPLES AND SCALES OF NOTATION. § 1. QUANTITY is the answer to the question quantus, how much? It is therefore that property of objects by means of which, when two of the same kind are compared together, one can be said to be greater or less than the other. § 2. Magnitude, which is often used as identical with Quantity, is really the answer to the question, how great? and may be used of everything which admits of the notion of greater or less, although in common language it is usually used with reference to the bulk of an object. §3. Unit or Unity is the name given to that quantity which is to be reckoned as one, when other quantities of the same kind are to be measured. § 4. Number is the relation of a quantity to its unit; the notion of number being suggested by successive repetitions of the individual unit. § 5. When men first began to count, they would count numbers of some particular thing: so many men, so many horses, &c. Next they would observe that whatever result is obtained, as by adding one number of men to another number of men, the same result would be true if the same numbers of any particular kind of thing were used; if 15 men and 3 men more made 18 men, and 15 horses and 3 horses more made 18 horses, generally 15 and 3 would make 18, whatever kind of thing was reckoned: and the idea of number abstracted from any particular kind of thing would thus be realized. B Hence we define concrete numbers to be those considered as belonging to some determinate species; abstract numbers to be those taken without reference to any particular species. Thus in 12 inches, and 12 pence, the 12 is a concrete number. But if we say 7 and 5 make 12, or 7 times 5 are 35, the numbers used are all abstract. And even if we say that a foot is 12 times as great as an inch, the number 12 is still abstract. § 6. We can now explain more fully the term unit: it is not itself one; but it is the magnitude which shall be represented by one in calculation. If all lengths be referred to the standard of an inch, all weights to the standard of a pound, all periods of time to the standard of a second, the inch would be called the unit of length, the pound the unit of weight, the second the unit of time: that is to say, the unit would be a length, or a weight, or a time. The symbol which represents the abstract conception of singleness as distinguished from multitude is 1, which is the unit of abstract arithmetic but all concrete quantities must have units of their own kind; and indeed anything may be unity for other things of its own kind; i.e. the unit is at first arbitrarily fixed on: the unit of length might be a foot, or a yard; the unit of weight might be an ounce, or a stone. § 7. Arithmetic (apıμηtiên, scilicet réxvn) is the art of numbering; and is usually taken to mean the science of expressing numbers by symbols, and of applying set rules to the different operations in which numbers are used. § 8. Notation is the art of expressing numbers by figures or symbols appropriated for that purpose. § 9. Numeration is generally applied to the converse process of expressing in words a number which is already expressed in symbols. § 10. To explain what is meant by a scale of Notation. By a scale of Notation is meant a systematic arrangement for facilitating the computation of large numbers. Instead of giving independent names to the whole series of natural numbers beginning from unity, which would make an unlimited and most embarrassing nomenclature, it is arranged that a certain number of units arbitrarily fixed upon shall be grouped into a class; and that the same number of these classes shall be taken to form a class of the next higher order; and that the same number of these higher classes |