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The Early Period of the Calculus of Variations

Paolo Freguglia and Mariano Giaquinta
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
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The history of the calculus of variations has itself a long history, which begins with a survey by Lagrange in 1806, followed by the historical study done by R. A. Woodhouse in1810; after that, there was Todhunter’s history of 1861 — and many more since.

Spanning the period 1696 to 1771, this latest addition to the literature starts with Fermat’s work on the principle of least time and discusses early attempts at isoperimetric problems together with Galileo’s study of the quickest descent curve. Subsequent emphasis is on Johan Bernoulli’s research on the brachistochrone problem, which marks the birth of the calculus of variations as a subject in its own right.

During this ‘early period of the calculus of variations’, the major contributors to the field were the Bernoulli brothers, Leonhard Euler and J. L. Lagrange. Euler’s contributions to real analysis enabled him to free the subject from reliance on geometric methods of the sort employed by the Bernoullis. Then, in 1762, Lagrange introduced an early version of his more powerful methods of the \(\delta\)-calculus, from which the ubiquitous Euler-Lagrange equations arose.

The first half of the book describes the achievements of the Bernoulli brothers on quickest descent curves, isoperimetric problems and geodesics on a surface. From this context, Johan Bernoulli’s mathematical prescience led him to the important fact that geodesics have osculating planes that are perpendicular to the surface. This was taken up by Clairaut, who applied it to the study of geodesics on surfaces of revolution. Between them, Euler and Lagrange built on the work of the Bernoullis and developed more powerful methods that were applicable to mechanics.

Freguglia and Giaquinta have provided much more than a timeline for the development of the calculus of variations. Most of the book consists of close examination of the major mathematical publications done by the aforementioned pioneers, and this is interwoven with many enlivening extracts from correspondence between such mathematicians.

Although the authors guide the reader through the mathematical writings of Bernoulli, Euler and Lagrange, such material is far from easy reading (why should it be?). Because early mathematics was expressed by means that are unfamiliar to us today, and since original writing was often of an exploratory nature, it requires specialist historians to unravel it for us. Moreover, even the great Euler was prone to errors of the sort highlighted at various points of this book.

Although addressed principally to historians, the authors regard the book as being of interest to students who may wish to follow the “step by step” progress of analysis in the 18th century. It is a reference work of great integrity, but not necessarily something that would be read in one sitting.

Peter Ruane is retired from the field of mathematics education, which involved the training of primary and secondary school teachers. His postgraduate study included algebraic topology and differential geometry, with applications to superconductivity.

See the table of contents in the publisher's webpage.