Letters

Leonurdo,Vol. 6,pp. 281-287. PergamonPress 1973. Printedin Great Britain. LETTERS Readers’ comments are welcomed on textspublished in Leonardo. TheEditors reserve the right to shorten lettersfor reasonsof space. Lettersshould be written in Englishor in French. MATHEMATICS OF GEOMETRY IN ABSTRACT PAINTINGS (cont.) I am concerned with some mathematical errors and misinterpretations that occurred in Crockett Johnson’s article in Leonardo 5, 97 (1972) and in the subsequent letter by Michael Holt in Leonardo 6, 91 (1973). Some of the points that I mention might seem rather pedantic but I feel that artists who proclaim that they use mathematical methods and terminology must use them precisely. A good example of the type of mistake that can occur with an inexact use of mathematics is given by Vantongerloo [l] where he falselyproves that ~ / C O = co andm x w = l . First, I will consider a few points from Johnson’s article. He states that ‘n... cannot have a finite value’. This is a misuse of the terms finite and infinite. n is most certainly a finitenumber because it is between the finite numbers 3 and 4, mathematically 3 < n <4. What Johnson means is that the decimalexpansion of n is non-terminating. Later in the article Johnson in discussing the accuracy of his ‘Squared Circle’ painting states that the error, that is the difference between the area of the square and the area of the circle, would not exceed a certain distance. But one cannot compare an area with a length, as they are measured in differentunits. Finally, I would like to ask Johnson why in his painting ‘Square Divided by Conic Rectangles with Gnomons Added at the Suggestion of a Paleolithic Colleague’ did he add a tilted square? In all the previous paintings that he describes all the lines appear because they are essential in the construction of the figure but in this painting, as far as I can understand, the tilted square is added only to make it look similar to an illustrated cave painting. What I would like to know is if Johnson has found any constructional reason for adding the tilted square. In the same issue of Leonardo 5, 169 (1972) the word ‘gnomon’ is defined in the Terminology section. While the given definition is correct, there is another meaning of the word that is more usual in mathematics and art. It is defined as the area added to a given area so that the resultant area is similar to the original area (Fig. 1). This definition is given in D’Arcy Thompson [2] and in W. and T. A. Millington [3]. Whilst I agree with most of the sentiments expressed in Holt’s subsequent letter, I am not happy with some of his mathematical ideas. He states the following: ‘The point about transcendentalnumbers , suchas K, is that the pattern of digits does not repeat itself.’ While this is true, it is also true for some non-transcendental numbers such as 2/2. Later he statesthat the transcendental number e=2.71 ... is the limit of the Golden Section but in his own book [4] he correctly states that the limit of the Golden Section is 1.618 .... Lastly, he states that ‘the pure mathematician will have little truck with numbers which are “nearly” anything, such as 42’. As a pure mathematician,I disagree with this because 2/2 is as exact as any other number and without the irrational numbers, of which 4 2 is one, a very large part of pure mathematicswould ceaseto exist. My main feeling about artists such as Crockett Johnson who work with ideas derived from Greek geometry is that they are missing so much. An enormous amount has happened since then and, though much of modern mathematics is incomprehensible to a layman, there are several fields that have developed in the last hundred years that can be understood by an amateur. Modern mathematics is exciting and a very powerful tool and, if artists can get to gripswith it, this will come over in their art. FIa. 1 REFERENCES 1. G. Vantongerloo,Paintings, Sculptures, Reflections (New York: Wittenborn, Schultz, 1948)p. 16. 281 282 Letters 2. D’Arcy Thompson, On Growth and Form (Cambridge , England: Cambridge University Press, 1966) p. 181. 3. W. andT. A. Millington,DictionaryofMathematics (London:Cassell, 1966)p. 101. 4. M. Holt, Mathematics in Art (London: Studio Vista, 1971) p. 62. John Wilson Dept. o f Mathematics University o f Wales,Institute o f Science and Technology King Edward VIZAve. CardiflCFl 3NU Wales, U.K. ON ENTROPY AND ART (cont.) Arnheim may indeed not have set out to ‘attack‘ anything in his book, though he certainly mounts some forceful attempts to discredit, particularly in his strictures on information theory and statistical thermodynamics (Leonardo 6, 188 (1973)).Let me amplify some of my remarks on the latter, leaving Peter Lloyd Jones to defend his own aesthetic flank. It is no part of my intention here to press for a gross and insensitive projection of physics into art theory-only to clarify below misconceptions and to insist that one’s licence to use physically inspired concepts at a metaphorical level should be conditional upon getting the physics right. 1. Arnheim’s misconception about statistical physics being restricted to non-interactive systems seems to me not to be an innocent slip, as is clear from the way he makes it bear the whole animus of gestalt theoretics against any tendency to view processes as the sum of individual particularseven , one might reasonably complain, as a first approximation. His suggestion that physicists are somehowin default on this is nonsenseand must be refuted. Modern theoretical physics is not simply concerned with but utterly dominated by the ‘many-body problem’, whose literature is permeated throughout with gestalt-reminiscent terminology , for example, collective modes, long-range order, cooperative phenomena and N-body forces. The reason why no one makes a fuss about ‘gestaltphysics ’ is partly that there are no methodological issues involved but also because this particular emphasiscouldonlybetruisticandofzeropredictive value. SeeingArnheim mislead himself in this way, it is difficultas a non-specialist to resist the impression that other psychologists may be right in regarding gestalt notions as non- or even counterproductivein their ownfield. 2. At one point in his book (p. 10) Arnheim poses the rhetorical question: ‘What is the relation of the two cosmic tendencies:towards mechanical disorder (entropy principle) and towards geometrical order (in crystals, molecules, organisms etc.)?’ He seems, by failure to provide a prompt answer, to suggest that the physical as well as the aesthetic problem is fraught with mystery and paradox. It ought to be stressed that, so far as everyday physics is concerned, an explicit answer to the question is available and can be found in any thermodynamics text. It takes the form: F (free-energy)=E (internal energy) -TS (temperature x entropy), coupled with the assertion, due to Gibbs, that for finite, open systems, which may gain or lose energy subjectto a fixed temperature, the free-energy function F is a minimum at equilibrium. This expresses precisely the appropriate compromise between E and S (Arnheim’s anabolic and catabolic tendencies, if one will) and replacesthe principle of maximum entropy appropriate to closed systems (or, problematically, the universe as a whole). Thus, two points arise. Some kind of restriction of the discussion to finite, open systems would seem to be necessary to obtain a useful formulation of the energy-entropy antithesis on the metaphorical plane,just as on the physical one. (Surely there is an echo here of the often admitted need for selectivity and discrimination in the act, if not the spirit of creation.) Secondly,as the above equation makes clear, anyone using ‘energy’ together with ‘entropy’ outside the physical context had better consider the implication of a metaphorical ‘temperature’as well. Use of this in a socio-aesthetic framework, while a shade prosaic, seems no more and no less objectionable or constructive than use of ‘entropy’. For what it is worth, non-physical temperature-like parameters have been used in cybernetics,theoreticaleconomics,demographyetc. These arejust two aspectsof a complexissuethat will undoubtedly provoke further discussion. Perhaps I should not have suggestedthat Arnheim’s wholeenterprisecollapsesbecauseof them-though I would not like the task of underpinning it once the more obvious misinterpretations and fallacies are removed. As I tried to indicate, I welcome and admire his coming into the open with a challenge that, for all its faults, is utterly uncontaminated by what he has elsewhere called ‘. .. the dazzling obscurityof arty talk, the juggling with catchwords and dehydrated aesthetic concepts, the pseudoscientificwindowdressing ,the impertinent hunting for clinical symptoms, the elaborate measurement of trifles, and the charming epigrams’. His correct discernment that many people are tired of these things might well serve as a spur for Leonardo’s criticalconscience. Michael Hoare 20 Bradbourne St. London SW6 3TE England Let this be enough. Only one more correction. The question about the two cosmic tendencies attributed to me by Michael Hoare (Books, Leonardo 6, 76 (1973))is a quotation from L. L. Whyte, clearly labeled by me as such. (What is happening to the art of reading?) And, of course, the question was not rhetorical. Whyte formulated it as one of several questions ‘which are already Letters 283 clear enough for us to be aware of our ignorance’. My own different formulation is given on p. 34 of my book. Rudolf Arnheim Dept. of Visualand Environmental Studies Harvard University Cambridge, MA 02138, U.S.A. With reference to Rudolf Arnheim’s letter in Leonardo 6, 188 (1973) on the technical point of quotation or, as it should be, dramatic irony, I am sorry my punctuation did not convey adequately my intention, which was to put a rhetorical question in order to summarize in one sentence three pages of Arnheim’s own difficult book. This phrase was of course mine. We obviously need a variety of graded quotation marks! My summary was, I think, a fair one (Leonardo 6,29 (1973)). The samepoint could be made to turn on a direct quotation such as one from p. 54 where Arnheim, in his book, discusses the responsibility for ‘diagnosing disorder where, to the best of our judgment, it prevails. Nor does it give us licence to accept disorder in the work of an artist as an interpretation of disorder when we recognise it as a mere addition toit’ (italicsmine). More generally, I would like to stress that Arnheim’s letter reinforces my point that value judgments shouldbe explicit. For instance,Warhol is attacked covertly with a sneering sideswipein the courseof a condescendingreferenceto my apparent approval of him. I tried to keep my artistic likes and dislikes out of my article altogether. As it happens I suspect that my artistic prejudices are pretty close to Arnheim’s. I doubt though whether his book has done their defence much of a service. Peter Lloyd Jones 22 Bradbourne St. Fulham,London S.W.6, England I remarked in 1961 about the entropy concept that, in spite of its age, ‘it has kept an untarnished lustre of novelty, an aura of unplumbed depth. It may well be that it holds further surprises in store for us’ [I]. I also said that it teaches us that knowledge is one and indivisible. The searchings and arguments about entropy in the pages of Leonardo seem, therefore, to me to be both exciting and instructivebut not (entirely) surprising. It is, after all, an ideal notion to bring together artists and scientists. As I said on the same occasion, the cooperation between them should be built on good foundations: ‘To the traffic of samples through a hatch (handed by the scientist to the arts man), there should be added a traffic of ideas, and to this latter traffic, I hope I have shown that the entropy concept could surely make a significant contribution .’ So my first reaction to what has been written in the pages of Leonardo is one of deep satisfaction that this particular topic has indeed sparked off a dialogue so that artists and scientists are at least talking the same sort of language and, in this case, about related things. Under these conditions, I do not mind too much if what is being said by an artist is partially wrong as science, or if a scientist in his criticism is occasionally a little extreme or, if I may say so, ungenerous. For a scientist should appreciate that artists and others have (in Arnheim’s book [2] and elsewhere) tried to take a purely scientijic notion, struggledwith it, expounded it and then attempted to use it. This seems excellent , and Arnheim has done just that, as is clear from his impressivelist of references. But of course Michael Hoare, in his penetrating review [3] is quite right that Arnheim has, in his book, misinterpretedscience, since the entropy of a system of interacting components (not only of independent components) is susceptible to experimental and theoretical study (p. 21). Arnheim has furthermore remarked (incorrectly) that disorder provides the maximum information (p. 19, whereas great information, in fact, is provided by a large and clear departure from order. Thus it is difficult to imagine a statement more pregnant with information than: ‘The Sun did not rise this morning!’ There is, of course, a great dualism underlying much of the discussion of the supposedly pessimistic aspect of entropy, i.e. of the eventual ‘heat death‘ of the universe. It is argued that the entropy concept has therefore a sombre and darkening effect on art. It seems to me that this is false scienceand poor analogy: 1. False science,becausethe entropy of universes that contract again after a certain maximal expansion is not yet fully understood. It could be that the entropy decreases in the contracting epochs and there could, therefore, be a rebirth and a new beginning. 2. False science, too, because the entropy in a volume of space can be alwaysdecreased, provided this volume of space is not fully insulated from its surroundings. The entropy will then increase elsewhere. 3. Poor analogy, because the (doubtful) notion of a heat death, which thermodynamicshas added to our culture,is surelyovershadowedby far greater sadnesses such as the passing of persons, eras and places; poverty; sickness;jealousy and war. In the face of these thoughts, what does it matter that the life of the Sun may be limited to a few million years? The second point surely clears up the puzzle referred to by various writers [1-4] that creative activity and biological structure formation are compatible with the second law of thermodynamics . The most recent contribution to the discussion by Gal-Or [5] is quite right in emphasizing that among scientists themselves various fundamental notions of thermodynamicsare expressed in widely differentways, soas to lead oneto suspectthat there are profound differences of opinion. This par- 284 Letters ticular point was, in fact, first made by me when I cited 19 distinct quotations concerning irreversibility from expert scientists [6]. These disagreements and nuances will, however, be cleared up as active research on problems of thermodynamic irreversibility proceeds and need not be a cause of worry to artists or scientists. Gal-Or’s concluding remark that science no longer needs the entropy concept is a private opinion that will not be shared by the many engineers, chemists and physicists who actually have to calculate entropy differences. A change of name from erttropy to energy dissipation, if it is only a change of name, would not of course affect anything discussed in this correspondence. 1. 2. 3. 4. 5. 6. REFERENCES P. T. Landsberg, Entropyand the UnityofKnowledge (Cardiff:Univ. of WalesPress, 1961). R. Arnheim, Entropyand Art: An Essay on Disorder and Order (Berkeley: Univ. of California Press, 1971). M. Hoare,Books,Leonardo 6,76 (1972). L. L. Whyte, Atomism, Structure and Form in Structure in Art and in Science, G. Kepes, ed. (London:StudioVista, 1965). B. Gal-Or, Letters,Leonardo 6, 188 (1973). P. T. Landsberg, Time in Statistical Physics and Special Relativity, Studium Generale 23, 1108 (1970). Reprinted in The Study of Time, J. T. Fraser, F. C. Haber and G. H. Muller, eds. (Berlin: Springer, 1972). See also International Conference on Thermodynamics,Cardiff1970. P. T. Landsberg, ed. (London: Butterworth,1970). P. T. Landsberg Dept. of Mathematics University of Southampton Southampton SO9 5NH England ON INFORMATION AVAILABLE IN PICTURES (cont.) I was glad to see the comments by Couzin ‘On Gibson’s and Goodman’s Accounts of Depiction’ (Leonardo 6, 233 (1973)). If the controversy over pictures, perspective and perception persists long enough, perhaps some of the confusions will be clarified. We have a long way to go, however, for I sometimes think that there is no problem in the history of human thought in such a deep intellectual mess as the problem of perspective. There are so many different ways of going wrong in thinking about it that confusion is excusable. And there are so many different disciplines that have to besorted outand taken intoaccount that anyonecan go wrong-optics, geometry, physics, philosophy, art history, opthalmology and psychology. It was because of this welter of disciplines that I thought I had to invent a new one, ecological optics. In the last analysis, I agree with what Couzin says and I am grateful for his support, since Nelson Goodman, like Erwin Panofsky before him, is a formidable opponent. But I think he gets himself into unnecessary complexities in discussing systems of perspective. He seems to confuse projection to any point in spacewith projection to a point of observation in a medium. The first only has to do with abstract projective geometry, the second has to do with the actual optic array to an eye. The diagram he has provided to illustrate inverted perspective does not have the eye at the point of projection. It is quite meaningless. What I really intended to say in my article (Leonardo 4, 27 (1971)),not to be polite about it, is that to speak of a system of inverted perspective is nonsense. When I refer to the information in either an optic array from a picture or an optic array from an environment, it is not easy to understand what I mean, unfortunately. And perhaps the concept of a formless invariant is obscure. But I assuredly do not mean what Couzin says I must mean: ‘Though Gibson does not seem to realize it, his account still requires that a picture must necessarily resemble the object depicted.’ Since picture and object share information, he says, ‘thisguarantees a likeness in the perceptions of both‘. But this is truly a misunderstanding. I want to maintain that a picture need not be related to its object by a point-projection of it. It is related to its object by invariant-projection. A good caricature then does not resemble the face of the person caricatured; it specifies him instead. The question of similarity or resemblance or likeness simply does not arise. James J. Gibson Psychological Research Laboratories Cornell Industrial Park Zthaca, N Y 14850, U.S.A. In my note (Leonardo 6,233 (1973)), I separated the question of ‘what is to count as a system of perspective’ from the question of which systems are adequate for faithful depiction. Gibson disagrees with my liberal answer to the first question, in particular with my considering a projective system where the projection point, instead of being the viewing point, is elsewhere on the line of sight. In fact, I do not require that the ‘rule of location’ of a system of perspective be either projective or in any way dependent on viewing point (or sense-of-view). In this way, nonstandard perspective systems, which are typically reactions against viewing point constructions, are not faulted immediately but laid bare and then subjected to the second question. The first question is a terminological one, to be answered strategically . The second question is the tough one, to be answered with an account of depiction. ‘Resemblance’has at least two poles of meaning: as a relation of similarity between things on the basis of their perceptible properties and as a relation between things on the basis of a relation of similarity between our perceptions of these things (on the basis of their properties). Gibson seems to be denying that his account requires resemblance in the first sense, while my argument is that it requires resemblance in the second sense. Resem- Letters 285 blance is an unruly notion and a major culprit in confusions about depiction. My descriptions of the two poles are not quite clear and what is between the poles is clamouring for attention. Dennis Couzin Dept. of Philosophy University of Illinois at Chicago Circle Chicago,IL 60680, U S A . ON BOOK REVIEWS ‘Secondary School Art’ In reviewingJ. Portchmouth‘s book in Leonardo 5, 276 (1972), I was so impressedwith its contents and flow that I decided to exclude any criticism that might detract from his unusually fine book, except for a loaded suggestion for further improvementto which the author responded in Letters, Leonardo 5, 384 (1972). Actually, I meant the suggestion more as a broad challenge to art educators in general to find ways and means to help students go all the way toward the unification of their paintings. To be sure, to implement this concept adds more problems for the instructor but for the student the benefits in heightened esthetic perception, sensitivity and art appreciation, from my experience, justifythe effort. Portchmouth’s awareness of student needs and his methodology for inspiring creative expression are of a top quality. Yet, to spur the student’s imagination without helping him to structure the results is not enough. As Dick Field says: ‘Each problem must be dealt with in the knowledge that there is to be finally a whole thing to which the solution will contribute’-Change in Art Education (London: Routledge and Kegan Paul, (1972) p. 34) (cf. Books, Leonardo 6, 276 (1973)). Students respond to the discipline of scanning the whole surface area to establish esthetic relationships of two-dimensional shapes, tonal values, textures, linesand colors. In justice to Portchmouth, he is concerned with the possibility of the student losing the character of the vision expressed. But in aiming no higher Portchmouth is, unwittingly, doing his students an injustice. He has but to envision the attainment of both aims: to strengthen the student’s particular kind of vision and to broaden the student’sinsight to art through inculcating the habit of structuring the whole. Strategies are available for instructors to accomplish these aims without interfering with the student’s personal vision. David Friend 945 Crane St., A Menlo Park, CA 94025, U.S.A. ‘The Future of Man’ David Smith grasps precisely the aim of my book when hestatesitspurposeas‘stimulatingwholesome young intelligent couples to have larger families’ (Books, Leonardo 6, 69 (1973)). But then he becomes lost, perhaps among the footnotes that trouble him or because he is not sure what intelligenceis . Early man (a cave-dwelling hunter) rose above all other creatures largely because he evolved upward in intelligence instead of in size, strength or speed. However, about the time agriculturewas developed, I believe he ceased to evolve further in average intelligence. As natural selection diminished and the number of humans increased with the increased food supply, proletariats appeared and grew until (when led by intelligent exploiters) they were able to kill off most of the upper classes of their own people, especially the more intelligent ones. As Karl Marx pointed out, the French Revolution was the first great example of this. My hope is that man will not deteriorate in average intelligence but instead will produce more and more good minds. Robert Graham Armorlite, Inc. P.O. Box 1038 Burbank, CA 91505, U.S.A. ‘Changing Art, Changing Man’ Ken Baynes’ review of my book (Leonardo 6, 175 (1973)) is the most perceptive to date and his criticism is well-founded. I find, however, that a replyis in order. A work of art is a replication of the human psyche and, a great work, speaking to the whole history of the past, changes the psyche of the observer in a more profound way than ordinary experience. The remarkable evolutionary changes from matter endowed with form to life endowed with form and sensitivityand then to man possessed of form, sensuousness,and self-consciousness,puts man at the top of a pyramid of increasing complexity and sensitivity. The further and deliberate extension of awareness can be viewed as a purpose for man’s future. Then the art museum becomes more than ever a part of a superior man-made environment, a spiritual gymnasium where sensitivities can be stretched and strengthened. Art is one element in the next development of man. But only one. A new social matrix is essential in order to institutionalize the effort to enlarge man’s biologicalequipment .Isuggestthat sinceacquisivityasthe essential purpose of life begins at long last to show signs of disintegration, men, weary of the contemporary purposelessness, can begin the search for a new center of purpose beyond faltering capitalism and Christianity, as well as present communism. They are likely to find this center in a dedicationto the further advancement of man along his evolutionary road. We are reaching the limits of what man can endure in the area of disintegrationand the search for a new center of belief will expand. It seems to me that the most likely objectivewill be that of replacing our present concern with 286 Letters acquisivity with a concern for enlarging our sensitivities. Society will devote its energies through art and scienceand other meanstobettering the species. The rewards of such social selection will go to creativity and human insight rather than to production, greed and ruthlessness. In this, art and the museum will not be the sole aids to survival . Society, as a whole could develop a social spirit and devotion like that which inspired the building of the great cathedrals-an odd phenomenon almost entirely without economic reward for those who paid the bills. Breakdown is apparent in societies with which I am familiar, if one looks at their art, which is full of madness, violence and sickness. It seemed a bit early when I wrote my book to go too far on the road to social change, which most people deeply and instinctively fear. But the rapid decay of the old paradigms raises the hope that the world is ready to hear of a new set of values. The most promising direction is toward devotion, through all social means, aesthetic, scientific, even economic , to make a system of rewards that will advance survivingman further along the road marked out by millions of years of past evolutions. Great fruitful changes can come out of such a devotion by man, once his magnificent energies are enlisted in this cause, so superior to the causes now being fought over. It is hard to imagine the possibility of such a change. But I am sure it is waiting for us and it will have more in it than just the art experience alone. David Mandel Counsellorat Law 313 State St. Perth Amboy, NJO8861, U.S.A. ‘Practical Carving’ I tend to the view that the purpose of a review is, firstly, to inform the reader on the nature and purpose of a book and, secondly, to examine critically its success or lack of success in terms of its purpose. However, as Nevelson belongs to the school that uses a review (Leonardo 6, 184 (1973)) as a peg to hang his prejudices on, may I be permitted to inform your readers as to the purpose and content of my book. It begins by discussing design and principles of carving, dealing with individual materials such as wood, stone and ivory, the tools required and their care and maintenance. My book is aimed at students, teachers, serious amateurs and artists who have little or no experience in this specific field. I am not an ecclesiastical carver nor an old sailor. If I were, I could not have written this book. On the contrary, I work as much with clay, plaster, resin and concrete as I do with stone and wood. But I have taught carving for many years and I am fully aware of the problems involved. I hold that to be an artist with an understanding of craft is the best qualificationfor teaching. And may heaven preserve us from critics, such as Nevelson, who believe that saleability in art galleries is the correct criterion by which to judge our work. Robert G. Dawson 2 Milton Park Highgate, London, N.6, England Dawson accuses me of the sin of prejudice but neglectsto clarify it. In my review (Books, Leonardo 6, 184 (1973)), I did not state, (or believe) as Dawson says, ‘saleability in art galleries is the correct criterion. ..’. I wrote: ‘In 1972, carving is not the vogue in cosmopolitan art galleries. Sadly, it is not commercially profitable.’ This is a matter of fact as anyone can easily observe. I wrote nothing to indicate a relationship between art galleries and quality of work. Dawson writes: ‘I hold that to be an artist with an understanding of craft is the best qualification for teaching.’ I hold that the best qualification for teaching is to be a professional teacher with a mastery of the subject being taught. Mike Nevelson 3 Milltown Road New Fairjeld, CT 06810, U.S.A. REQUEST FOR INFORMATION In connection with a research project, I am seeking information on British painters who visited America to work or teach between 1945-1970. I would be interested especially to hear from anyone who has suchinformation coveringthe period of the late ’40s and early ’50s. Mary Michaels Faculty of Art and Design ThePolytechnic Wolverhampton, North Street Wolverhampton WV1 1DT, England CORRECTIONS I regret that the following two errors occurred in my article ‘Perspective and the Viewpoint’ (Leonardo 5, 209 (1972)) in paragraph 2.2.8 on page 211: 1. The viewpoint is only restricted as described if one utilizes the assumption implied in Fig. 1 that the diagonals of ABCD are orthogonal in picturespace . However, as stated in paragraph 2.2.9, it was not intended to make that assumption in this context and, if one does not make it, then the viewpoint becomes indeterminate on MLN. 2. Fig. 1 and not Fig. 2 represents a schema often exploited in Baroque paintings. What I described as the standard schema in most 14th to 19th century European paintings is not illustrated (as a result of a revision of the manuscript) but it is Letters 287 described in paragraph 2.2.2. The schema was first givenexactformulationin the 15thcentury. 6, 77 (1973)), Bragdon's name has been misspelled and the book was publishedby Unicorn Bookshop, 50 Gloucester Road, Brighton, Sussex, England. Kenneth R. Adams 19 DartmouthPark Road London N.W.5, England Please note that in Anthony Hill's review of Projective Ornament by Claude Bragdon (Leonard0 Bill Butler UnicornBookshop 50 GloucesterRoad Brighton, Sussex, England ...