APPLICATIONS Since the applications of arithmetic reach every branch of trade, industry, commercial activity and science, it is evident that an arbitrary limit must be placed upon what the schools can offer. It is also evident that, so far as the subject matter is con cerned, this limit must vary with the world's needs and business customs. A further boundary is fixed by the abilities of the pupils in the elementary schools, this being a fairly measurable constant, at least in selected geographical and racial areas.
Each business application found in a textbook on arithmetic represents what is, or was at one time, a need. The problem of the pipes filling a cistern was a useful one in the days when the Romans established public fountains in every important town; partnership involving time was once as important as any problem involving corporations at the present time ; the problem of the couriers was a real one when communication from place to place depended upon human endurance and fidelity; and the banker's draft was known even in Babylonian times. The schools always tend to be conservative.
The study of child psychology, which had its scientific begin ning in the latter part of the 19th century, has established certain rather definite limits as to the nature of the arithmetic offered from year to year. This study does not concern the nature of problems so much as it does the pupil's ability to do the compu tation and to master the reasoning involved ; it relates to the ap plied problem to the extent of seeing that its meaning is fairly within the mental grasp of the child. This phase of the subject is at present attracting more attention than the actual practic ability of the applied problems. It has resulted in an improvement in the course of study and has assisted in eliminating certain cases of computation (as in the division of unusable fractions and decimals) that are beyond the needs and the ready grasp of ele mentary pupils. The result has not been the standardization of topics or of applications, but the fixing of limits of difficulty for pupils of various degrees of ability. The chief problem of the schools is to find, from the wide range of applications, those that the great majority of people will sometime need to use and that psychology shows are within the mental reach of the pupils at the time they are introduced. In each of these respects the schools are making progress that is definite and is probably as rapid as circumstances permit. Among the present applications which have replaced those which are now obsolete are the following: personal and household accounts, budgeting, household inventory, sales slips, bills and invoices, transportation problems, pay-rolls, corn munity problems (how money is raised and how it is spent), thrift and investments, business graphs, and government income and expenses.
Older books: T. Leslie, Philosophy of Arithmetic (1817, 2nd ed. 182o) ; D. Lardner, Treatise on Arithmetic (1834) ; A. de Morgan, Elements of Arithmetic (5th ed. 1846).
History: D. E. Smith, History of Mathematics vol. ii. (Boston, 1923, 1925) ; J. Tropfke, Geschichte der Elementar-Mathematik, vol. i. (2nd ed. 1921). (D. E. S.) Early Arithmetics: Cuthbert Tonstall, De ante supputandi (London. 1522) ; Robert Recorde, The Ground of Artes (London, 1542) ; Michael Stifel, Arithmetica Integra (Nurnberg, ; Gaspard de Texeda, Suma de Arithmetica pratica (Valladolid, 1546) ; Humphrey Baker, The Well Spring of Sciences (London, 1568) ; Jean Trenchant, L'Arithmetique (Lyons, 1578). These books are described in Rara Arithmetica, a catalogue of early Arithmetics, with an account of those in the library of George Arthur Plimpton, New York.