Summary — W. L. Crum and Joseph A. Schumpeter, Rudimentary Mathematics for Economists and Statisticians (1946)

Crum and Schumpeter’s textbook is a deliberately elementary initiation into the mathematical habits needed by economists and statisticians. It is not a treatise on mathematical economics, but a disciplined primer in functions, graphs, limits, derivatives, extrema, differential equations, and determinants, always tied to economic examples.

The objective of this book is to present rudimentary ideas and operations essential to any effective mathematical reasoning by economists and statisticians.

The book’s guiding claim is that the economist’s obstacle is often not advanced technique but weak command of elementary concepts: variable, function, coordinate, slope, rate, and symbolic formulation. Its opening chapters therefore use cost functions to move patiently among tables, diagrams, verbal assumptions, and equations. Total cost, average cost, fixed cost, overhead, and proportional cost become instruments for teaching how economic statements acquire precision when translated into mathematical form.

The desired mathematical representation takes two forms, the graphic and the symbolic.

This translation is not presented as decorative formalism. The authors repeatedly show that once an economic assumption is expressed as a function, its consequences can be found by algebraic and graphic reasoning. Constant average cost, fixed total cost, declining average cost, and U-shaped cost curves are treated as relations whose meaning depends on exact specification. The method is especially attentive to the danger of verbal ambiguity: what seems economically obvious may become mathematically false, incomplete, or unexpectedly rich when written as an equation.

To this end, a succession of particular assumptions concerning costs is needed; these assumptions are of the a priori sort—it is assumed that the law relating cost to quantity is known, without experimentation.

The middle chapters build toward calculus by making marginal concepts rigorous. Marginal cost, marginal utility, and marginal revenue are not treated as loose references to “small changes,” but as limiting ratios. Limits thus provide the bridge between elementary graphing and differential calculus. Once the derivative has been introduced, familiar economic problems can be restated as questions about rates of change, tangents, and extrema.

The derivative of $y$ with respect to $x$ is the instantaneous rate of change of $y$ with $x$.

The calculus chapters remain practical rather than ornamental. Rules of differentiation are introduced after their meaning has been established, and optimization is tied to economic interpretation: minimum average cost, maximum utility, least-cost production, and regression problems all become cases of locating and classifying extrema. The treatment of constrained maximization and the Lagrange multiplier similarly presents mathematics as a concise language for marginal adjustment under limiting conditions.

The later chapters reverse the direction of inquiry. Differential equations begin from marginal or rate relations and recover the underlying total function through integration; compound interest, depreciation, elasticity, and utility illustrate how economic laws may first appear in derivative form. The final chapter on determinants extends the same logic from single equations to systems, stressing that economic magnitudes are often mutually determined rather than isolated.

Historically, the book belongs to the moment when mathematical economics was becoming normalized while still requiring elementary exposition for many readers. Its lasting interest lies in its pedagogy: graphs discipline intuition, symbols clarify assumptions, limits define marginal magnitudes, derivatives analyze rates and optima, integrals reconstruct totals, and determinants organize simultaneous relations. The result is a compact apprenticeship in mathematical economic reasoning rather than a specialized technical monograph.