Gerhard Tintner’s Review of Rudimentary Mathematics for Economists and Statisticians (1946)

Gerhard Tintner’s book review assesses W. L. Crum and Joseph A. Schumpeter’s elementary mathematical textbook through a tension between accessibility and completeness. He welcomes an introduction that teaches economists through problems in their own discipline, while questioning the omission of proofs and the limited attention to statistics. His final judgment distinguishes mathematical breadth from teaching suitability: more comprehensive alternatives may be preferable in content, yet less appropriate for students with weaker mathematical preparation.

The review first surveys the book’s seven-chapter progression. Graphic analysis, analytical geometry, and elementary algebra begin with cost theory; limits are introduced through marginal cost, demand, marginal utility, and marginal revenue. The treatment then advances to differentiation, including higher, partial, and total derivatives and Taylor series. Maxima, minima, and inflection points receive applications from cost and production theory and linear regression. Differential equations and integration lead to elasticity, compound interest, total utility, and elementary economic dynamics, before a final introduction to determinants. This survey establishes both the range of material and the consistent use of economic examples.

The authors ought to be congratulated for having achieved almost a miracle of condensation.

Tintner’s praise identifies the book’s principal achievement, but also introduces its central pedagogical difficulty. Covering so many demanding topics in little space depends on stating mathematical theorems without demonstrating them. Compression is therefore not an uncomplicated virtue: it makes the subject accessible in scope and length while potentially making its reasoning harder to follow.

This lack of proof may, however, be confusing to the reader.

The criticism is specific rather than a demand for exhaustive rigor. Tintner argues that introducing the binomial theorem would have allowed an exact proof of the rule for differentiating a power. Its inclusion would also have served students of statistics and probability. He similarly maintains that other differentiation rules could have been proved without substantially enlarging the book. These observations qualify the apparent necessity of the trade-off between brevity and demonstration: at least some explanatory omissions were avoidable.

The statistical criticism extends beyond the absence of one theorem. Tintner finds that, despite the textbook’s intended audience, its only statistical example concerns simple regression. The extensive use of economics thus coexists with an uneven treatment of the two disciplines named in the title. His concern is not merely that additional topics would improve coverage, but that the chosen mathematical foundations and examples insufficiently address the statistician’s needs.

It is particularly recommended because it is free from the preoccupation with physics and especially classical mechanics which characterizes almost all introductory calculus books.

Here the disciplinary orientation becomes a decisive advantage. For Tintner, economists need not spend time mastering physical applications merely to acquire useful calculus. Economic examples reduce unnecessary preparatory burdens and connect mathematical techniques directly to their intended use. His recommendation is nevertheless directed toward the economist already somewhat interested and inclined toward mathematics, rather than offered as a claim that the book suits every beginner.

The concluding comparison sharpens this distinction between substantive excellence and classroom fit. Tintner regards R. G. D. Allen’s Mathematical Analysis for Economists as superior in mathematical range and its survey of traditional mathematical economics; he also somewhat prefers the textbook by D. C. Jones and G. W. Daniels. Yet both presuppose the stronger secondary-school mathematical education he attributes to English students.

Hence, it seems that they are not as suitable for teaching purposes as the present book.

The review’s qualified endorsement rests on this attention to readers’ preparation. Crum and Schumpeter’s textbook succeeds as a compact, economically oriented introduction, even though its brevity sometimes sacrifices intelligibility and its statistical provision remains thin. Tintner’s core evaluative move is to judge mathematical instruction by both the quality of its explanation and its suitability for the students who will actually use it.