The Geometry of An Art: The History of the Mathematical Theory of Perspective from Alberti to Monge

It is obvious (to me) that the postmodernist notion that meaning is ascribed not by the author but by the reader was born in the mind of an author reading a review of one of his or her own books. In bluntly personal terms — since I am English not French — no reviewer ever seems to have read the book I thought I had written, and I can only apologize for inflicting a similar experience on Professor Andersen.

The book will be widely read by people interested in the mathematics of perspective. And by and large it deserves to be. However, my advice to readers is to start in the chronological middle (about 1600) and refer back to the earlier chapters only if you feel you need to. This is, of course, seemingly perverse advice for a mathematical book (though it receives some support from the author, see below), and one of the difficulties I find with this volume is indeed the perceived relationship between a mathematical treatment of a subject and a historical one. But there is much of interest on the way.

The main title of the volume, The Geometry of an Art, does not really describe the contents, which have very little to do with any of the visual arts — understandably, since there is so much mathematics to deal with. The "an art" to which the title refers seems to be painting, which is to ignore the important and very extensive use of perspective by architects. (1) The subtitle, The History of the Mathematical Theory of Perspective from Alberti to Monge, is much more helpful, but is subverted by the title of Chapter VI (p. 237), "The Birth of the Mathematical Theory of Perspective: Guidobaldo and Stevin". The relevant dates are: Leon Battista Alberti (1404–1472), Gaspard Monge (1746–1818), Guidobaldo del Monte (1545–1607) and Simon Stevin (1548–1620). "The mathematical theory of perspective" that is the subject of the volume allegedly first sees the light in Guidobaldo's Six books on perspective (Perspetivae libri sex) of 1600. I have no quarrel with this, given what is here understood as a (or the) mathematical theory of perspective. But that understanding raises a mathematico-historical problem, one that is not explicitly addressed: Is it possible to have one definition that is valid for all the works considered?

Andersen has chosen to unify the mathematics. That is, to use the same notation, and the same lettering of diagrams, throughout her book. This makes the work less useful as a guide to the primary sources, but allows Andersen to tell a coherent story of logical development of mathematics. The historical problem, as I see it, is that the story is not coherent and can only be made to look that way by treating mathematics as a closed system isolated from social forces such as changes in taste in the visual arts, the fashion for perspective stage sets, and so on. The weaknesses of this 'internalist' approach have been apparent for several decades in the history of science, and 'internalism' is now largely unacceptable. So from the point of view of a historian of science, the onus is on Andersen to show it can be made to work in this case.

Moreover, the use of one set of technical terms in discussions of texts from the fifteenth to the eighteenth century, while obviously helpful for explaining mathematics, offends against the commitment to using "actors' categories" that is now the norm among historians of science, who have found that confining themselves to vocabulary that would have made sense to the people they are talking about is a powerful method of avoiding anachronism. To a large extent, Andersen has chosen to impose a modern mathematical scheme upon a disparate set of writings.

Andersen's choices help in making comparisons, but can be seriously misleading, particularly in the earlier, prenatal, phase. For instance, in connection with a proof by Piero della Francesca (c. 1412–1492) she writes

Having given this analysis of Piero's proof, I hasten to add that I do not think it should be judged by modern standards. It took a long time for the property of the vanishing point to be understood mathematically. Before this understanding was achieved, Piero's argument that SiTi has the right length could be seen as, if not a final proof, then at least as (sic) a strong support for the procedure used in his Alberti construction.

In Piero's proof, the point Andersen calls a "vanishing point" (a term introduced in 1715) is the meet of only two lines, and his proof depends only upon simple geometrical optics (from Euclid) and the use of similar triangles. The proof is perfectly correct (in Euclidean geometry). (2) This proof is fundamental to Piero's work, since it establishes the correctness of his perspective construction (which is not identical with Alberti's). The suggestion that the proof was incorrect originated with an art historian (1987) who had simply misunderstood Piero's text, having (for a start) failed to read the preliminary definitions. Instead of working through Piero's characteristically fifteenth-century proof, Andersen has complicated matters and somehow confused herself into accepting that the proof is incorrect.

A compendious rather than critical use of secondary sources is rather noticeable in this part of Andersen's book. In particular, I cannot understand her apparent respect for Morris Kline's Mathematics in Western Culture (London, 1954, second edition 1972), which was never a serious contribution to historical scholarship. Kline's opinions may lie behind the naïve assertion that Piero della Francesca's perspective treatise exercised little influence, partly because it was not printed (p. 79). There is abundant evidence that the treatise was read in manuscript; and Piero's examples, his construction procedures and some traces of his actual mathematics appear in all subsequent practical treatises on perspective. (3)

Things brighten up considerably when we emerge from the intrauterine practical tradition, to which Andersen seems to be basically unsympathetic, into the neonatal phase of mathematical theory. But there are lingering consequences of her neglect of a wider context and her modernisation of vocabulary, the most high-profile example being the strange assessment of the work of Girard Desargues (1591–1661). I entirely agree that his 12-page book on perspective (1636), which consists of one worked example, is not in itself of mathematical importance, though it does provide an early specimen of an "abbreviated" method, that is one that does not require the use of points outside the picture field. Moreover, Andersen presents an interesting account of the mathematical aspects of the squabbling that attended the use of Desargues' method in teaching at the Royal Academy of Painting and Sculpture in Paris in the 1650s. (4) However, Desargues is remembered not for perspective but for inventing projective geometry, which is formally equivalent to perspective, in a book on conic sections that was published in 1639. Andersen ignores the formal equivalence, the fact that the perspective book refers to treating parallel lines in the same way as convergent ones and to conic sections, and the fact that the book on conics refers to perspective. On the basis of there being no trace of cross ratios in the perspective work (and ignoring the fact that Desargues does not use cross ratios in his projective geometry) she asserts there is no connection between the two treatises, preferring to see them as evidence of his having a dual mathematical personality, one part allied to the practical perspective tradition and the other to the learned Greek geometry from which he drew projective geometry (we are not told how, or why it should happen in the early seventeenth century). This is all done in a page or so (pp. 446–447). Tougher argument is required if we are to consider dismissing what historians normally regard as the most important contribution of perspective studies to the development of mathematics. I think the cavalier style springs from Andersen's lack of intellectual engagement with the practical texts. She is much happier with texts written by researchers for researchers. And she is very interested in vanishing points.

Which is to say that Andersen is in her element when the narrative reaches the eighteenth century. This part of the book is the one with most authority. But even here the poverty of internalism is exposed. For instance, there is a substantial chapter on the elegant mathematics of Johann Heinrich Lambert (1728–1777) (Chapter XII, pp. 635–705) — I feel that this volume is a fat book with a healthier one about Lambert trying to get out — but in the final chapter (Chapter XIV) we are told "Lambert came farthest in relating perspective to other disciplines" (p. 718). This is to ignore the fact that Desargues founded a whole new way of doing geometry upon projection. Andersen may wish to deny that there is a historical link between his invention of projective geometry and his study of perspective, but she cannot deny the mathematical equivalence. The work of Desargues established a connection whether Desargues knew it or not. Moreover, Andersen's statement ignores the long established connections with geometrical optics as the basis of fifteenth- and sixteenth-century work in perspective. The perceived divergences from both the geometry of sight and an emerging psychology seem to have been responsible for changing attitudes to perspective from the mid seventeenth century onwards. These changes belong in the world of the sciences and the visual arts, a world that has been studied by art historians, most notably in two important books by Michael Baxandall and Martin Kemp, (5) and is touched upon in many other publications, several of which appear in Andersen's extensive bibliography of secondary sources (pp. 771–793). Andersen perhaps does not need to recap such work in detail but she should not have ignored it in drawing her conclusions.

The admirably clear analytical Contents pages (pp. vii–xviii), which detail the progress of the narrative, grouping authors according to national languages, provide a helpful supplement to the index (pp. 795–809), which may be the work of a professional indexer rather than the author, and is much less analytical (for instance, it gives a simple list of about fifty page numbers after the name Guidobaldo del Monte). As the Contents pages show, care has been taken to include all possibly substantial contributors to the mathematical story; and their dates have been included in the index. Thus the work will be useful for reference purposes. The corresponding danger, not always avoided, is that the style becomes that of a chronicle ("one damn thing after another").

This is reinforced by the Introduction's coy refusal to tell us where the story is going (p. xxiii: "I do not wish to reveal any of my conclusions at this premature stage"). The actual conclusion is almost equally inconclusive. The book ends with a chapter entitled "Summing Up" (Chapter XIV, pp. 713–21), which merely gives a summary of the developments described in preceding chapters. There is no overall conclusion except that we are told (p. 721) that "The forces that drove them [the protagonists] were their interest in and enthusiasm for the theoretical aspects of perspective, and their inclination to give in to the seduction of mathematics." The book is thus not presented as telling a specific story of its own or as recounting an episode in a larger story. This seems curiously weak for so substantial a piece of work, in which there has been much of mathematical interest. I am still enough of a mathematician to enjoy large tracts of this volume, but as a historian I have some serious misgivings.

The publishers, though not the author, richly deserve to have this review end on a sour note. Springer's copyediting is a disgrace. There is scarcely a page that does not contain an error or an infelicity, ranging from misuse of the definite article, via incorrect syntax, to comic turns such as "drawers" where the text requires "draftsmen" (p. 720). This kind of negligence should make potential Springer authors look to their contracts.

NOTES

1. See, for example, Filippo Camerota, La prospettiva del Rinascimento: Arte, architettura, scienza (Architetti e architetture no 19), Milan: Electa, 2006. Earlier work by this author is in Andersen's bibliography.

2. See J. V. Field,, The Invention of Infinity: Mathematics and Art in the Renaissance, Oxford: Oxford University Press, 1997 (in Andersen's bibliography); and idem, Piero della Francesca: A mathematician's art, New Haven and London: Yale University Press, 2005 (not available to Andersen when her book was being written).

3. See, for example, M. J. Kemp, 'Piero and the Idiots: The Early Fortuna of his Theories of Perspective', in M.A. Lavin (ed.), Piero della Francesca and His Legacy, Washington, D.C.: National Gallery of Art (Studies in the History of Art, no 48, Center for Advanced Study in the Visual Arts, Symposium Papers XXVIII), 1995, 199–211 (in Andersen's bibliography); and Lyle Massey (ed.), The Treatise on Perspective: Published and Unpublished, Washington D. C.: National Gallery of Art (Studies in the History of Art, no 59, Center for Advanced Study in the Visual Arts, Symposium Papers XXXVI), 2003 (not in Andersen's bibliography).

4. On other aspects, see M. J. Kemp, ' "A chaos of intelligence": Leonardo's 'Traité' and the perspective wars in the Académie royale', in André Chastel et al., 'Il se rendit en Italie': Etudes offertes à André Chastel, Rome: Edizioni dell'Elefante, 1987, pp. 415–426 (in Andersen's bibliography); reprinted in Claire Farago (ed.), Re-Reading Leonardo: The Treatise on Painting across Europe from 1550 to 1900, Aldershot: Ashgate, in press.

5. Michael Baxandall, Patterns of Intention: On the Historical Explanation of Pictures, New Haven and London: Yale University Press, 1985 (not in Andersen's bibliography); M. J. Kemp, The Science of Art: Optical themes in western art from Brunelleschi to Seurat, New Haven and London: Yale University Press, 1990 (in Andersen's bibliography).

J. V. Field, who is an Honorary Visiting Research Fellow at the School of History of Art, Film and Visual Media of Birkbeck, University of London, is the author of Kepler's Geometrical Cosmology, The Geometrical Work of Girard Desargues (with J. J. Gray),  The Invention of Infinity: Mathematics and Art in the Renaissance and Piero della Francesca: A Mathematician's Art.