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The statistical operation of multiple linear regression by least squares is equivalent to the orthogonal projection of vectors of observations on a space spanned by vectors of observations; and a partial regression can be similarly represented as an oblique projection. This connexion between a statistical and a formal algebraical operation gives the main source of interest for this investigation. Its object is to develop algebraical theory which supplies terms necessary for a unified algebraical and geometrical formulation of concepts in multivariate analysis. An exposition of the statistical theory is to be made in a separate paper.

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In the investigation of preference orders which are explanations of expenditure data which associates a quantity vector xr with a price vector pr (r = 1, …, k), in respect to some n goods, there is considered the class of functions φ, with gradient g, which are increasing and convex in some convex region containing the points xr, such that gr = g(xr) has the direction of pr. † Let ur = pr/er, where so that

This book approaches various aspects of economics that have to do with choice, and the opportunity for it, such as individual and social choice, production, optimal programming, and the market. The topics belong mostly to microeconomics, but they also have other connections. The object is to state a view about choice and value and to give an account of the logical apparatus. With this there is a wish to present limited matters fairly completely and unproblematically, and where there is some issue about their nature to consider that also. The book consists of six parts, each containing several chapters. Parts I–IV deal with generalities about choice, individual or social, and representative economic topics. The remaining parts have more concern with straightforwardly mathematical subjects, which have an application or interpretation for economics but need not be exclusively connected there. Chapters often are fairly self‐contained or belong to sequences that can be taken more or less on their own. The topics are in the main fabric of economic theory, and most students encounter them. A preamble at the start of every chapter tells what it is about; from this and possibly some further scanning, the main ideas should be easily gathered by those who might not be concerned with all details. Expository materials and reworkings of published fragments have been joined with unpublished work from past and recent years. In all there is a view about choice and ‘the optimum’ in economics surely acceptable to some and perhaps what they have always thought but undoubtedly not to everyone.

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