is to stand is defined, so the graph attains its greatest efficiency as a symbol only when the nature of the functions which are to be represented is clearly specified in advance, as well as the properties of functions which are to be represented by special features of the curve. As has been seen above, the characteristics of first and second derivatives seem to be particularly adapted to graphical representation, and it has been suggested that curves possess their fullest significance as symbols of functions when the functions are continuous, have only a finite number of maxima and minima in any given interval, and have continuous derivatives of the first and second orders. The elementary functions have these properties, in common with all of the other functions which have been designated as analytic. But it is not necessary that the functions represented be thus restricted in character, provided only that the correspondence between the analytical characteristics of the function on the one hand and the graphical characteristics of the curve on the other, is expressly understood. In the elementary courses it is evidently impossible to discuss the niceties of the relation of graphical to analytical conceptions, and it is highly desirable that graphical methods should be used. But they should always be formulated with special reference in the mind of the instructor to the correspondence between the graphical and the analytical processes, with which the student will later be familiar. We have now come to the end of our brief survey of the elements of the calculus, the threshold of the higher mathematics. The technical difficulties which would arise have prevented the application of the processes of differentation and integration to any but the simplest functions, the polynomials. By means of these alone, however, it has been possible to explain the meaning of the derivative, the anti-derivative, and the definite integral, and some of their interrelations among themselves. The rest of the theory is for the most part an application in many different ways and to many different functions of these three fundamental conceptions. It is hoped that by his perusal of these pages the reader unfamiliar with 304 with the calculus will have lost whatever awe he may have had of one at least of the more advanced mathematical subjects, and at the same time have gained an insight into the variety and importance of its relations with problems of a practical nature and with other branches of science. 32, Theoretic solution of the linear congruence in one unknown; 33, Numerical solution of the linear congruence in one unknown; VII THE THEORY OF NUMBERS By J. W. A. YOUNG I. INTRODUCTION 1. The "Theory of Numbers" might, in a certain sense, include nearly all of the subject-matter usually treated in mathematics, since, with the exception of the non-metrical portions of Geometry, there are few domains of mathematics that are not fundamentally concerned with numbers. But the term is commonly used in a restricted, technical sense as meaning the theory of integral numbers (positive, negative, zero). Even this must be further restricted, for all numbers other than integers can be defined in terms of integers,* so that to study the whole body of theory that has been built up on integral numbers would still be tantamount to studying nearly the whole body of mathematical science. The restriction customarily made is to regard the theory of numbers " as concerned with integers as such; their properties and their combinations by operations that lead to integral results. The operations of addition, subtraction, and multiplication are accordingly admitted when applied to any integers, and division is admitted when applied to integers such that the quotient is integral. The process of division may also be used to obtain equations between integers. For example, 9385-62.151 +23.† In all that follows the term number shall accordingly be understood to mean integral number; and other terms, for * See Monograph IV, Appendix I. 307 |