A History of Geometrical Methods

First published by Oxford University Press in 1940 and then reissued in the Dover series in 1963, this book had been out of print for many years until this recent reprint in the Dover Phoenix edition. Its return is a most welcome event since it is unique in the scope of its coverage and its style of analysis.

Whilst there are many histories of mathematics, and various histories of particular geometrical topics, there is no other single publication that examines the historical development of the methods by which geometry has grown into the vast subject that it had become by the time the book was originally published (sixty three years ago). Even at that time, the author declared the nature of the task that he had set himself to be rather overwhelming.

As if to get an early grip on his subject, Coolidge begins by describing the geometrical abilities of spiders in the manufacture of webs and he mentions the expertise of honey-bees with respect to maxima-minima problems. All of this is referenced to research findings by a variety of writers on these topics. Ensuing pages are devoted to 'early geometry' of the Babylonians and the Egyptians, followed by a look at contributions from China, Japan and India.

The contents of the book are organised under three main headings:

1. Synthetic geometry, meaning the geometry of all the predecessors of Descartes followed by synthetic approaches to projective geometry, non-euclidean geometries and descriptive geometry.

2. Algebraic geometry, beginning with the historical emergence of the use of coordinates and algebraic equations connecting them, proceeding to the use of groups of transformations up to the birational group.

3. Differential geometry, from the work of early writers such as Euler and Monge proceeding to intrinsic geometry, the Gaussian theory of surfaces, projective and absolute differential geometry.

The omission of the obvious candidate for a fourth category, Topology, was made for two very valid reasons. Firstly, Coolidge confessed his unfamiliarity with this rapidly growing branch of mathematics and, secondly, its relative newness was a bar to any sort of objective over-view. Nonetheless, the book is very substantial as it stands and extends to over four hundred pages.

The three categories above are not treated equally in terms of space devoted to them by Coolidge. Synthetic geometry is covered in about the first one hundred pages; algebraic geometry attracts the greatest attention with about half the text devoted to it (200 pages) and a historical discussion of the methods of differential geometry fills the last quarter of the book.

To clarify the meaning of the term 'geometrical methods', I quote some of those examined by the author under each of his three categories of content (above).

For synthetic geometry of the Greeks, there are the methods of construction (compasses and straightedge), the use of deductive methods via axioms and postulates, Archimedes' methods of extension and equilibrium, Eudoxus' method of exhaustion and so on. Then, for later elementary geometry, there is discussion of further work on methods of construction as, for example, the problem of Apollonius, the trisection of angle, duplication of the cube and circle constructions of Mohr and Mascheroni and consideration of Poncelet's constructions with ruler and circle.

Computational approaches are also studied, beginning with the formulae of Hero and Brahmagupta, culminating with a look at some of the work of Isaac Barrow.

Also under the banner of synthetic geometry, the chapter on projective geometry begins with a description of the work of Desargues and culminates with discussion of the methods used by Steiner and von Staudt, although there is very little mention of the axiomatic basis. There is also reference to early Greek mathematicians and their apparent awareness of the projective invariance of cross ratio and, in particular, the fact that Menelaus knew that this certainly applied in the case of spherical geometry.

The early history of algebraic geometry commences with discussion of the methods used by Fermat and Descartes and proceeds to study those used by Newton, McLaurin etc and going as far as Monge and Euler.

There are six subsequent chapters on the methods of algebraic geometry described as follows:

• Systems of linear coordinates

• Point coordinates

• Enumerative geometry

• Birational geometry

• Higher spaces and higher space elements

• Geometrical transformations.

The last of these examines the use of projective transformations, circular transformations, Cremona transformations, the Kleinian viewpoint and Lie theory.

Covering all of this in fifty pages means that the treatment is somewhat condensed, but the historical thread emerges with clarity.

The final part of the book deals with differential geometry under five chapter headings:

• Early writers (Newton to Cauchy)

• Intrinsic geometry (Euler's use of evolutes to the idea of Cartan's moving frame)

• Gauss and the classical theory of surfaces

• Projective differential geometry

• Absolute differential geometry (Riemannian and non-Riemannian).

Naturally, the emphasis is upon the classical approach to differential geometry, with virtually no mention of the use of differential forms and manifolds.

Obviously, some parts of the book need to be updated with respect to various historical judgements. For example, Coolidge does not clarify the origins of vector methods, although readers may infer that Hermann Grassmann was the founder. Neither is it made clear that he was the inventor of the concepts of inner and outer products. Clarification of this is, of course, provided by Martin J. Crow [1] in 1967. Readers may also be interested to compare Coolidge's coverage of the history of the methods of algebraic geometry with that provided by Jean Dieudonné [2].

Overall, this book is a remarkable achievement and, despite the large amount of subsequent research into the history of mathematics since its publication, it should still occupy a prominent place in the in the libraries of mathematical historians. It is mathematically challenging and, with about six hundred sources quoted in the index of authors, it is also very well researched from a historical perspective. The subject index, however, is somewhat inadequate.

References:

[1] A History of Vector Analysis, M. J. Crowe. (Dover, 1985)

[2] "The Historical Development of Algebraic Geometry," J. Dieudonné. (American Mathematical Monthly, vol 79, 1972, pp. 827-866)

Peter Ruane (ruane.p@blueyonder.co.uk) was Senior Lecturer in Mathematics Education at Anglia Polytechnic University, England. His research interests lie within the field of mathematics education and the history of geometry.