# Researches into the Mathematical Principles of the Theory of Wealth/Preface

PREFACE
The science known as Political Economy, which for a century has so much interested thinkers, is to-day more generally diffused than ever before. It shares with politics proper the attention of the great journals, which are to-day the most important means of spreading information; but the public is so tired of theories and systems that now the demand is for so-called "positive" matter, i.e. in political economy, custom-house abstracts, statistical documents, and government reports, such as will throw the light of experience on the important questions which are being agitated before the country, and which so greatly interest all classes of society.
I make no objection to this tendency; it is good, and in accord with the laws which govern the development of all branches of science. I will only observe that theory ought not to be confounded with systems, although in the infancy of all sciences the instinct of system necessarily attempts to outline theories. I will add that theory should always have some part, small though it may be, in the development of a science; and that, to a man of my profession in particular, more than to any other, it should be permissible to consider from an exclusively theoretical ​standpoint, a subject of general interest which has so many different sides.
But the title of this work sets forth not only theoretical researches; it shows also that I intend to apply to them the forms and symbols of mathematical analysis. This is a plan likely, I confess, to draw on me at the outset the condemnation of theorists of repute. With one accord they have set themselves against the use of mathematical forms, and it will doubtless be difficult to overcome to-day a prejudice which thinkers, like Smith and other more modern writers, have contributed to strengthen. The reasons for this prejudice seem to be, on the one hand, the false point of view from which theory has been regarded by the small number of those who have thought of applying mathematics to it; and, on the other hand, the false notion which has been formed of this analysis by men otherwise judicious and well versed in the subject of Political Economy, but to whom the mathematical sciences are unfamiliar.
The attempts which have been made in this direction have remained very little known, and I have been able to learn only the titles of them, except one, Les Principes de l'Économie Politique, by Canard a small work published in the year X [of the French Republic, A.D. 1801], and crowned by the Institut. These pretended principles are so radically at fault, and the application of them is so erroneous, that the approval of a distinguished body of men was unable to preserve the work from oblivion. It is easy to see why essays of this nature should not incline such economists as Say and Ricardo to algebra.
I have said that most authors who have devoted ​themselves to political economy seem also to have had a wrong idea of the nature of the applications of mathematical analysis to the theory of wealth. They imagined that the use of symbols and formulas could only lead to numerical calculations, and as it was clearly perceived that the subject was not suited to such a numerical determination of values by means of theory alone, the conclusion was drawn that the mathematical apparatus, if not liable to lead to erroneous results, was at least idle and pedantic. But those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression. Thus the theory of probabilities furnishes a demonstration of very important propositions, although, without the help of experience, it is impossible to give numerical values for contingent events, except in questions of mere curiosity, such as arise from certain games of chance. Thus, also, theoretical Mechanics furnishes to practical Mechanics general theorems of most useful application, although in almost all cases recourse to experience is necessary for the numerical results which practice requires.
The employment of mathematical symbols is perfectly natural when the relations between magnitudes are under discussion; and even if they are not rigorously necessary, it would hardly be reasonable to reject them, because they are not equally familiar to all readers and because they have sometimes been wrongly used, if they are able to facilitate the exposition of problems, to render it more concise, to ​open the way to more extended developments, and to avoid the digressions of vague argumentation.
There are authors, like Smith and Say, who, in writing on Political Economy, have preserved all the beauties of a purely literary style; but there are others, like Ricardo, who, when treating the most abstract questions, or when seeking great accuracy, have not been able to avoid algebra, and have only disguised it under arithmetical calculations of tiresome length. Any one who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.
I propose to show in this essay that the solution of the general questions which arise from the theory of wealth, depends essentially not on elementary algebra, but on that branch of analysis which comprises arbitrary functions, which are merely restricted to satisfying certain conditions. As only very simple conditions will be considered, the first principles of the differential and integral calculus suffice for understanding this little treatise. Also, although I fear that it may appear too abstruse to most people who have a liking for these topics, I hardly dare to hope that it will deserve the attention of professional mathematicians, except as they may discover in it the germ of questions more worthy of their powers.
But there is a large class of men, and, thanks to a famous school, especially in France, who, after thorough mathematical training, have directed their attention to applications of those sciences which particularly interest society. Theories of the wealth of the community must attract their attention; and in considering them they are sure to feel, as ​I have felt, the need of rendering determinate by symbols familiar to them, an analysis which is generally indeterminate and often obscure, in authors who have thought fit to confine themselves to the resources of ordinary language. In thinking that they may be led by their reflexions to enter upon this path, I hope that my book may be of some use to them, and may lessen their labour.
In the remarks on the first notions of competition and the mutual relations of producers, they may possibly notice certain relations, which are very curious from a purely abstract standpoint, without reference to proposed applications.
I have not set out to make a complete and dogmatic treatise on Political Economy; I have put aside questions, to which mathematical analysis cannot apply, and those which seem to me entirely cleared up already. I have assumed that this book will only fall into the hands of readers who are familiar with what is found in the most ordinary books on these topics.
I am far from having thought of writing in support of any system, and from joining the banners of any party; I believe that there is an immense step in passing from theory to governmental applications; I believe that theory loses none of its value in thus remaining preserved from contact with impassioned polemics; and I believe, if this essay is of any practical value, it will be chiefly in making clear how far we are from being able to solve, with full knowledge of the case, a multitude of questions which are boldly decided every day.