Dear ReaderI venture to discard the more stately forms of preface which alone are considered suitable for a serious work, and to address a few words of direct appeal to you.An enthusiastic but candid friend, to whom showed these pages in proof, dwelt in glowing terms on the pleasure and pro t that my reader would derive from them, if only he survived the first cold plunge into `functions.' " Another equally candid friend to whom I reported the remark exclaimed, Survive it indeed! Why, what on earth is to induce him to take it?"Much counsel was offered me as to the best method of inducing him to take this cold plunge," the substance of which counsel may be found at the beginning of the poems of Lucretius and Tasso, who have given such exquisite expression to the theory of sugaring the pill" which their works illustrate. But I am no Lucretius, and have no power, even had I the desire to disguise the fact that a firm grasp of the elementary truths of Political Economy cannot be got without the same kind of severe and sustained mental application which is necessary in all other serious studies.At the same time I am aware that forty pages of almost unbroken mathematics may seem to many readers a most unnecessary introduction to Economics, and it is impossible that the beginner should see their bearing upon the subject until he has mastered and applied them. Some impatience, therefore, may naturally be expected. To remove this impatience, I can but express my own profound conviction that the beginner who has mastered this mathematical introduction will have solved, before he knows that he has even met them, some of the most crucial problems of Political Economy on which the foremost Economists have disputed unavailingly for generations for lack of applying the mathematical method. A glance at the Index of Illustrations" will show that my object is to bring Economics down from the clouds and make the study throw light on our daily doings and experiences, as well as on the great commercial and industrial machinery of the world. But in order to get this light some mathematical knowledge is needed, which it would be difficult to pick out of the standard treatises as it is wanted. This knowledge I have tried to collect and render accessible to those who dropped their mathematics when they left school, but are still willing to take the trouble to master a plain statement, even if it involves the use of mathematical symbols. The portions of the book printed in the smaller type should be omitted on a first reading. They generally deal either with difficult portions of the subject that are best postponed till the reader has some idea of the general drift of what he is doing, or else with objections that will probably not present themselves at first, and are better not dealt with till they rise naturally.The student is strongly recommended to consult the Summary of Definitions and Propositions on pp. 142{{144 at frequent intervals while reading the text.P. H. W.