V. Short There are numerous short processes of performing operations, or rather of securing results by substituting simpler operations than those to be performed. Thus to multiply by it is often easier to annex two zeros (or move the decimal point two places to the right) and divide by 8. In the same way it is easier to multiply by 100 and divide by 4 than to multiply by 25. Such processes depend upon simple number relations of the following kmd• 121= W, 25 = 331= 125 =--- 75%4, 12591%, 66170=1. The publication of extensive tables and the perfecting of calcu lating machines (q.v.) have rendered obsolete most of the short processes involving other kinds of multipliers and divisors.
VI. Compound four funda mental processes with compound numbers were formerly considered of much importance, since before the introduction of decimal fractions most tables of denominate numbers were on a varying scale. Within a century, however, the metric system (q.v.) and various monetary tables have so decimalized denominate numbers as to take from compound numbers most of their former importance. The only case in which several denominations are commonly used in writing a number to-day is that of English money. In most countries the whole subject is obsolete. The United States still uses the British system except in the monetary table, but it has greatly simplified it, rarely using more than two denominations in the same number. Indeed, within a single generation the metric system has come to be used exclusively in this country in scientific laboratories, and the efforts now being made to secure a large foreign trade will make the system more and more known in commercial and industrial affairs.
VII. Methods of Solving Problems.— There are five general methods of attacking an applied problem, as follows: (1) We may study typical problems and thus acquire the habit of solving others of the same nature. This is the oldest method, and was practically the only one in use before the 17th century. At present it is coming into renewed prominence in American schools, the type problem being attended (as was not formerly the case) by a large number of exercises.
(2) We may commit to memory rules for all general classes of problems liable to be met. Historically, this is the second method of at tack, and it characterizes the American text books until nearly the close of the 19th century. The rules were usually inductively inferred from type problems, and pupils committed them to memory. Since in practical life we never depend upon a verbatim rule, this method is rapidly becoming obsolete. In medieval times there was much effort expended in searching for a general rule that would solve all arith metical problems. Hence arose the Rule of
Three (see ARITHMETIC, HISTORY or), the Rule of False Position, and other rules of less im portance, all of which lost their chief value when algebraic symbolism was invented. Of these general rules only the Rule of Three has survived, being now recognized in the form of proportion.
(3) We may learn formulas instead of rules. This method was received with some favor for a time, but it has been discarded as a general plan. It has all of the defects of the method of rules, with the added difficulty of an unnecessarily confusing algebraic symbolism.
(4) We may analyze each problem as it arises, simply applying common sense to the solution. When problems are, as they always should be, properly graded to the understand ing of the pupils, this plan is better than any of 'the preceding ones. It establishes a habit of independence and of confidence that is wholly wanting in the older methods.
(5) We may bring to the aid of analysis the representation of the unknown quantity by the familiar algebraic symbol .r. This mate rially simplifies the analysis, and most writers on arithmetic at the present time advocate the plan. The concept of the linear equation with one unknown quantity is a very simple one, and it greatly clarifies the analysis in many cases.
VIII. Nature of the Problems in ArithViii. Nature of the Problems in Arith- metic.— The interests of the ancient and mediaeval philosophers were not at all commer cial. These men were attracted rather by con siderations of the properties of numbers and by puzzles which were imagined to sharpen the wit. The rise of commerce in the later Middle Ages and at the time of the Renaissance, brought into the science a large number of applied problems representing actual business conditions. Principles of conservatism have tended to keep these ancient problems from generation to generation, strengthened by the feeling that mental discipline was as well secured from an obsolete as from a modern problem. It is therefore only recently that the question has arisen: What should be the nature of the problems set for children study ing arithmetic? In answer to this question teachers seem to be tending to observe the fol lowing principles: (1) A problem that pretends to set forth a business custom should state the real business conditions of the present. This excludes obso lete business problems, it being the opinion that better mental discipline can be secured from a question relating to genuine commercial matters of the present, than from one relating solely to forgotten customs.