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(a)  Precedents: Charles Henry and Peter Lenz 

Kahnweiler in his book on Gris is anxious to stamp hard on any suggestion that Cubism (which is to say, for him, Picasso, Braque, Léger, Gris) could be understood as a mathematical art and, in particular, any notion that Picasso, Braque and Gris might have been influenced by the ideas of their friend Maurice Princet, who had a lively interest in new - non-Euclidean - gometrical systems (p.185). This view has been challenged in Linda Dalrymple Henderson's highly influential The Fourth Dimension and Non-Euclidean Geometry in Modern Art, a study that draws on an enormous wealth of documentation to illustrate the importance of its theme throughout the history of Cubism and of the earliest abstract art. Henderson's book represents a substantial opening up of the real history of early twentieth century art, Cubism included. In her discussion of the 'Salon Cubists', however, she maintains the division - still virtually unquestioned at the time she was writing - between 'analytical' and 'synthetic' Cubism and continues to treat the work of the Salon Cubists as contingent on that of Picasso and Braque. In the historiography of Cubism it is therefore, to use a terminology favoured by Gleizes, a reform rather than a revolution. 

There was of course nothing new in the Cubists' interest in mathematics. A connection between mathematics and painting had already been established by the Neo-Impressionists, Seurat and Signac, and by the Nabi, Paul Sérusier. Although Sérusier's interest in the Golden Section and the root rectangles was entirely Euclidean in nature, (31) Seurat and Signac were associated with Charles Henry who was arguing for the scientific idealism that seems to me the most important notion that needs to be retained in the present discussion.  That is, to recapitulate, the view - or rather recognition of the fact - that whether or not the world exists independently of consciousness it is only as a phenomenon of consciousness that it can be known.

31   I discuss Sérusier and his interest in geometry in my Afterword to Desiderius Lenz: The Aesthetic of Beuron.

Henry was an adept of the science of 'psycho-physics' proposed by the German Gustav Fechner, which argues that physics is nothing other than the study of mental processes. It consequently attached more importance than classical or practical physics to aesthetics, refusing Locke's distinction between primary and secondary qualities and insisting that the effect of colours and form on the sensibility was as much a reality, and deserved as much attention, as studies of mass, distance and velocity. The psycho-physicists also insisted that these were measurable, that aesthetic pleasure could be translated into numbers, that certain proportions were pleasing (or dynamogène) and others displeasing (or inhibitoire). Henri Bergson's Essai sur les données immédiates de la conscience is largely an attack on this idea that subjective experience is susceptible to measurement and though Charles Henry is not named it seems reasonable to assume that Bergson has him in mind, the more so since an outright polemic against Henry was launched on similar grounds by the very Bergson influenced Georges Sorel. (32) Straightaway we may remark that one of the themes running through On "Cubism" is a distinction between what is measurable and what is immeasurable.

32   Sorel: Psycho-physical Contributions to Studies in Aesthetics  [Contributions psycho-physiques à l'étude esthétique] and Aesthetics and Psycho-physics [Esthétique et psychophysique].

Henry was to play a role in the later history of Cubism, exercising great personal influence on both Gino Severini and Albert Gleizes. (33) I have found no signs of any involvement in the earlier history of Cubism but it is difficult to believe that it did not interest him.

33   This is discussed in my Albert Gleizes: For and Against the Twentieth Century and the introductions to Gleizes: Art and Religion etc and Gleizes/Severini: From Cubism to Classicism etc. 

Metzinger tells us in Cubism was Born that even at Nantes, prior to his arrival in Paris, he had already come to the conclusion, in opposition to his academically minded teacher, Hippolyte Touront, that the secret of an art able to resist the ravages of time, lay in number and proportion. It was the precise measurement of the red stockings in the painting ascribed to Murillo that gave it its strength. It was on the basis of this conviction that he turned to the Neo-Impressionists. He claims to have painted in the Neo-Impressionist manner solely on the basis of what he had read about them prior to ever having seen one of their paintings. 

We may also note Paul Sérusier's passing comment, 'I am the father of Cubism'. (34) Kahnweiler, ridiculing the assertion as we might expect, points to the fact that one of his pupils at the Académie Ranson was Roger de la Fresnaye (Juan Gris, p.184). Henderson claims (Fourth Dimension, p.61) that la Fresnaye formed the link between the circle of Alexandre Mercereau (Gleizes, Metzinger and Le Fauconnier) and the Société Normande de la Peinture Moderne, which included the Duchamp brothers who were to play a particularly important role in the history of the relations between mathematics and Cubism.

34   As quoted in e.g. Escholier: La Peinture française, XX siècle, p.14.


(b) The subjective experience of space 

None of this brings us into the territory of Non-Euclidean geometry and the Fourth dimension but I think it establishes a useful starting point for the discussion. The idea is already well established that there is a relation between the human sensibility and numbers and that mathematical formulae can be helpful in creating an effect that is unrelated to the associations, pleasurable or otherwise, of the painting's subject matter. Metzinger's Neo-Impressionist paintings are distinguished (like those of his friend at the time, Robert Delaunay) by the fact that he uses much larger dots of colour and these may be organised into clearly delimited blocks bound by what amount to being curves, arabesques and straight lines in a way that suggests Seurat's own last paintings and Signac's Portrait of Félix Fénéon, both heavily influenced by Henry. (35) He then goes into a still highly coloured, rather Gauguin-like, arabesque Fauvism in which it is tempting to see the influence of Sérusier; and it is from there that he jumps, rather startlingly (as we shall see, it startled his friend Apollinaire (36)), to his first Cubist paintings, exhibited in the Salon d'Automne in 1910 and showing a very obvious influence of the near contemporary work of Picasso and Braque. At much the same time he wrote the Note on Painting, which groups Picasso, Braque, Delaunay and Le Fauconnier together as representatives of a common school. It is, then, in the Note on Painting that we might reasonably hope to have some idea of the reasons for his dissatisfaction with his own earlier work. In it, he says: 

'Cézanne showed us how forms live in the reality of light. Picasso brings us a material account of their real life as it is lived in the mind; he lays the basis for a free, mobile pespective out of which the judicious mathematician Maurice Princet has deduced a whole geometry.'

35   See the analysis in Homer: Seurat and the Science of Painting.

36   See the section on Picasso and Braque below. 

'Their real life as it is lived in the mind.' Much later, writing in the Afterword to the 1947 edition of On "Cubism" he says: 

'It is no longer enough to look at the model, the painter must think it. He brings it into a space that is at once spiritual and plastic, about which it is not entirely frivolous to talk about the fourth dimension. There, proportions become qualities; sensations are no longer tied to a rigid system of axes and it is their expressive value alone that determines the order in which they are shown. The situation of the different parts of a figure, a still life, a landscape, no longer depends on that of the other parts; it depends on their situation in the mind of the artist, on their true situation ... Outside science and its instruments, the object - a group of sensations - can only be seized in its entirety by memory or by desire. It is to the representation of the internal reality, the only one that counts from the point of view of art, that Cubism is attached.' 

And this echoes much of the language of On "Cubism" itself, for example: 

'He [Courbet] had no notion of the fact that it is only through the operation of thought that the visible world becomes the real world, and that the objects that strike us most forcefully are not always those that are the most rich in truths of a plastic nature'

And again: 

'To establish pictorial space, we must turn to tactile and motor sensations, to all our faculties. It is our whole personality which, contracting or expanding, will transform the plane surface of the painting. As, in response, this plane will reflect this personality onto the understanding of the person looking at it, we can define pictorial space as follows: an exchange, accessible to the senses, between two spaces that are themselves subjective.'

What is meant by 'tactile and motor sensations' is explained in Cubist technique

'Perspective obliges us to use a trapezoid to depict the square facade of a house if we are looking at it from an angle. We have to forget everything we have learned through a long experience of movement and of touch under the supervision of the reason.' 

So, simply, it is not only through immediate vision that we know what shape things are but also through our daily experience of movement and touch. It is, to use an example given later in the essay, by turning it in our hand that we know the orange has a uniform colour, and by feeling it that we know it is a globe. The artist has a perfect right to make use of that knowledge. 

Henderson uses the reference to 'tactile and motor sensations' to point to the influence of Henri Poincaré, whose Science and hypothesis contains a chapter on 'Space and Geometry' which discusses, together with non-Euclidean geometry and the fourth dimension, 'tactile and motor space'. But I think we must be careful to avoid giving the impression that Metzinger, or any of the other painters with the possible exception (I shall discuss this further on) of Marcel Duchamp, are attempting to follow a formula, or put into practise an idea they have discovered in Poincaré. On the contrary. They are dissatisfied with the conventional laws of painting, the need to fit everything into an illusion of a three dimensional box, because it inhibits them from expressing certain qualities that interest them. These are essentially plastic qualities, even though Metzinger will insist (and Gleizes would presumably agree with him at this time) that they derive from thoughts and feelings excited by contact with objects found in the external world. Hence the rejection of abstraction, which the painters identify with mere decoration, and the insistence that their art is 'realist'. 

It is realist but the reality in question lies in the mind and in the subjective experience of the artist who, being an artist, is probably interested in plastic qualities. Out of this whole 'group of sensations' - which include tactile and motor sensations - the artist will choose those aspects that can be expressed on the flat surface of the canvas, and out of them a 'total image' will appear: 

'Picasso does not deny the object, he illuminates it with his intelligence and with his sensibility. To perceptions that are visual, he adds perceptions that are tactile. He undergoes an experience, he comes to an understanding of it, he organises it. The painting that results will be neither a transposition nor a schematic representation. In it we will contemplate the equivalent, made accessible to the senses and brought to life, of an idea - the total image.' (A Note on Painting) 

It is obvious that such a conception is incompatible with conventional perspective. Perspective, which is itself a geometrical formula, (37) imposes certain conditions on the painter which seriously inhibited this process of thinking the object through. To quote Cubism was Born

'Exaggerating reliefs and depths, the painters struggled to create an absurd, theatrical space in which two sides of a pathway would come together and prevent the traveller from going any further; the circular opening of a vase was reduced to a simple straight line; and a blessing was given to all the imperfections of our visual mechanism, all with the infantile aim of adding a supplementary dimension to what, since the time of the original Chaos, only possesses two.' 

The ambition to produce the object as it is thought and not as it is seen necessarily implied a different space: in fact it implied a different space for each painting. In Cubism was Born, referring to Gleizes, Metzinger says he had 'never heard Maurice Princet construct an infinite number of different spaces for the use of painters ...' And in On "Cubism", there is a very interesting formula - very interesting for the future development of Gleizes - that the form creates its own space: 'The Cubist painters are aware of this, those who tirelessly study pictorial form and the space to which it gives rise.' 

The form is not something that exists in a neutral element called 'space'. Every picture (indeed every plastic experience of any sort, including walking round a room) causes us to experience space differently and, since space is itself nothing other than a subjective experience, each of these spaces is different, each has its own laws. And the possibilities are infinite. 

Did the painters learn this from the study of geometrical textbooks? It would be equally possible to say that scientists such as Helmholtz and Henry learned it from the artists (of different sorts. The whole line of thought passes through Goethe). The most important thing to retain - and the point is stressed in On "Cubism" - is that the painters are not copying any formula devised by the geometers. But we can easily imagine the excitement they must have felt when they realised that their own dissatisfaction with classical perspective - and their conviction that the reality of the world lay in their own experience of it, and that this was much richer than could be conveyed by the conventional means of description and copying appearances - was shared by many scientists. And that each of the two sides had a lot to learn from the other. (38)

38   The argument would be developed much later by Gleizes in his Art and Science (1933).


(c) Metzinger, Gris and Maurice Princet 

None of this is to suggest that Metzinger at least did not see the new geometries as having implications for the actual practice of painting. I would suggest, against the suggestions of Henderson and of Arthur Miller (in his Einsten, Picasso - Space, Time and the Beauty that causes Havoc) that Picasso's interest was largely philosophical rather than practical. He would have seen the analogy and it would have reinforced his conviction that he was on the right track, but (despite Metzinger's remark that: 'As for Picasso, the specialist [Princet] was amazed by the rapidity of his understanding. The tradition he came from had prepared him better than ours for a problem to do with structure.' (39)) it is difficult to imagine him using real mathematical calculations to establish the proportions and overall construction of his paintings. It is much easier to see mathematical calculation in the beautiful, strong lines of construction we find in Metzinger.

39   Cubism was Born 

Herschel Chipp says that: 'At one time, Metzinger and Gris underwent a study of geometry under the direction of Princet in order to explore these possibilities.' (40) He does not give a source for this but we may assume he had it from an interview Metzinger gave him in 1952. It would be interesting to know when this 'study of geometry' occurred. There is a particularly intense artistic relationship between Metzinger and Gris which lasted through the period of the war but which has been occulted by the accounts of Kahnweiler and his epigones. (41)

40   Chipp: Theories of Modern Art, p.223.

41   To my knowledge Christopher Green was the first historian to hint at it (Cubism and Its Enemies, pp.25-37 on 'Crystal Cubism').  His Juan Gris, p. 29 suggests an influence - which I don't myself see - of Tea-time on Gris' Homage to Picasso. A much fuller account could be put together from Gris' correspondence with Léonce Rosenberg published by Christian Derouet (eg pp.34, for Feb 1915; 36, Sept 1916; pp.45-6 and 65 for joint ventures between Gris, Metzinger and Cocteau in 1917; p.78 for Metzinger visiting Gris in Beaulieu, July 1918; and pp.80-81 for very interesting thoughts on the relation between the two painters from Rosenberg himself who knew both of them well and saw Metzinger as very much the senior partner though at the time of writing he was feeling aggrieved with Gris who had abandoned him to return to Kahnweiler). Gleizes frequently links the two names but the connection is fairly obvious from a comparison of the paintings.

In another letter written by Metzinger in Paris to Gleizes in Barcelona during the war (4/7/16), he says: 

'After two years of researches I have succeeded in establishing the foundations of this new perspective about which I've talked to you so much. It is not the materialist perspective of Gris nor metaphysical perspective - I take responsibility for the word. You cannot imagine how much I've worked since [the start of] the war, working outside painting but for painting. The geometry of the fourth space has no more secret for me. Previously I only had intuitions, now I have certainties. I have made a whole series of theorems on the laws of displacement [déplacement], of reversal [retournement] etc. I have read Schoute, Rieman (sic), Argand, Schlegel (42) etc.

42   Schoute, Pieter Hendrik (1846-1923) Danish mathematician who worked on mutidimensional geometry; Riemann, Georg Friedrich Bernhard (1826-66) Mathematician whose name is particularly associated with a geometry in which the plane surface is treated as spherical, thus altering Euclid's postulates that parallel lines will never meet and that a line can be extended indefinitely in both directions; Schlegel, Victor (1843-1905) Mathematician who devoted much of his career to championing the work of Hermann Gunther Grassmann, pioneer of vector analysis, an attempt to develop a mathematical account both of position and of direction.

'The practical result? A new harmony. Do not take the word harmony in its commonplace [banal] meaning, take it in its original [primitif] meaning. Nothing is anything other than numbers. The mind [esprit] hates the immeasurable, it has to be reduced. That is the secret. Nothing is left over once the operation is completed [pas de reste à l'opération]. Painting, sculpture, music, architecture, lasting art is never anything other than a mathematical expression of the relations that exist between the internal and the external, the self [le moi] and the world.' 

The importance of this letter for the history of Cubism in Paris during the war cannot be exaggerated, but that lies outside the scope of the present study, which is concerned with On "Cubism" and its associations. There are, however, several points that should be retained. First, the association/rivalry with Gris (which we have also seen in the letter of 26th July 1916 quoted above). Secondly, the fact that Metzinger claims to have been working directly on mathematics - 'I have made a whole series of theorems'. Thirdly the emphasis on mathematics as the link between the mind and the world as it exists outside the mind. And fourthly, that everything in art is measurement: L'esprit haït l'incommensurable - the mind hates the immeasurable. That in Metzinger's view is what art is - the reduction of the intolerable chaos of existence to the harmony of numbers. 

The last point is especially important because it contradicts the argument of On "Cubism", which actually celebrates an aspect of art that is 'incommensurable': 

'we must act in such a way that no two parts with the same extension should find themselves together in the painting. Good sense approves of this and explains it: if one part repeats another part, the whole becomes measurable, and the work ceases to be a means of giving our personality (which is not susceptible to measurement, since nothing in it is ever repeated) a permanent form [une fixation de notre personnalité].'

We may, I think, reasonably speculate that this emphasis on the immeasurable comes from Gleizes and that it marks a longstanding disagreement between the two painters over the usefulness of these mathematical researches. In the year following his receipt of this letter, Gleizes wrote, in Modern Painting

'we spoke of the sterility to which art would be led by dangerous adventures [incursions] in the squaring of the circle, or in the mathematical absolute of a Henri Poincaré. Already before their birth, which we could tell was coming, we were chary of the dogmas, the hermeticisms, destructions disguised under the mask of a new construction.' 

Gleizes is responding to the fashion for the new geometries that was surrounding him among the admirers of Marcel Duchamp in New York but he must have been aware that he was distorting Metzinger's thought - or else trying to recall him to an earlier position he thought had been agreed between them. The most relevant passage in On "Cubism" has often been quoted: 

'The Cubist painters are aware of this, those who tirelessly study pictorial form and the space to which it gives rise. 

'We have fallen negligently into the habit of confusing this space either with pure visual space or with Euclidean space. 

'Euclid in one of his postulates asserts that figures do not change their shape when put into motion - which spares us the need to say anything more about that. 

'If anyone wanted to attach the painters' space to any sort of geometry they would have to turn to the non-Euclidean specialists, to reflect on certain of the theorems of Riemann.' 

Which seems to suggest that the painters do not base themselves on any sort of geometry but if they did, it would have to be non-Euclidean. 

Metzinger elaborates on this theme in Cubist Technique when he says: 

'I admit that Cubist perspective does touch upon certain geometrical expressions that official science likes to characterise as Utopian, and I am not even far from believing in the existence of an artistic geometry which is still only in a germinal stage. This, however, has no bearing on the state of the Cubist technique as it is at the present time.'  

Which complements the letters of 1916. Cubism in 1912 is not based on any specific geometry but Metzinger has the feeling it could be and that non-Euclidean geometry could provide it. By 1916 he believes he has found it. 

Nonetheless I would suggest that if Metzinger and Gris did set themselves to studying under the direction of Princet, it was before On "Cubism" was written and probably before 1912, when Gris gave up his work as a satirical cartoonist to concentrate full time on painting. This is speculation on my part and my evidence is admittedly tenuous. 

In a passage already quoted in Cubism was Born, Metzinger says: 'Albert Gleizes did not know Montmartre, had never seen anything of Picasso or Juan Gris, never heard Maurice Princet construct an infinite number of different spaces for the use of painters ...' 

This implies that Metzinger saw Gris as a significant player in Cubism prior to his own friendship with Gleizes, which began in 1910 and continued through 1911 and was therefore prior to Gris emerging as a painter in his own right. In the Salon d'Automne of 1910, Metzinger exhibits work that looks like a copy of the 'analytical Cubism' of Picasso and Braque. Much the same could be said of the Nude he exhibits in the Salon des Indépendants in 1911. It is unfortunate that these paintings are lost (43) but they appear from the photographs to be characterised by a Picasso/Braque like spatial ambiguity. The same could not, however be said for Tea-time, exhibited in the Salon d'Automne in 1912. Here, everything is clear, precise and measurable. The whole is inscribed in a beautifully constructed armature of straight lines and curves whose relation to each other is not determined by the figuration (the woman enjoying her tea) but interweaves with it in a manner that is entirely intelligible. We can see clearly how the lines interact with each other.

43   According to the catalogue of the 1912 Section d'Or exhibition, most of them seem to have passed through the hands of members of the Gleizes family - his mother, 'M.G.Comerre', and his uncle, Honoré Auclair, and 'Robert Gleizes', whose precise relationship to Gleizes has still not been established. Note, however, fn 59.

John Richardson, no friend of Metzinger, tells us, without giving a source, that this was the picture that persuaded Gris of the importance of numbers in painting.  (44) Gris starts painting seriously in 1911, and first exhibits in the Salon des Indépendants in 1912. He appears with two styles. In one of them a grid structure appears that is clearly reminiscent of the Goûter and of Metzinger's later work in 1912. In the other, the grid is still present but the lines are not stated and their continuity is broken. Their presence is suggested by the heavy, often triangular, shading of the angles between them (the illustration to On "Cubism" is an example as is the portrait of Picasso). Both styles are distinguished from the work of Picasso and Braque by their clear, rational and measurable quality.

44   Richardson: Picasso, p.211. Whatever the source is it might explain Christopher Green's view (see fn 41 above) that Tea Time had influenced Gris' Homage to Picasso.

Given that Princet lived into his nineties and amassed a small fortune  (45) without changing his job as actuary for the L'Abeille Insurance Company, it is surprising how little we know about him. At some point - but I am not in a position to say when - he seems to have disappeared completely from the history of painting. One thing we do know however is that he ridiculed the manner in which perspective reduces, say, the opening of a cup, which we all know to be a circle, to an ellipse. He taught that the full circle should be shown. We have this from an interview with André Lhote in 1952, (46) but a related argument (on painting what we know as opposed to what we see) features prominently in the account of Cubism in Gino Severini's From Cubism to Classicism (47) in a passage which I take to be largely a critique of Metzinger. It is also posed in an essay by Gleizes - Painting and Descriptive Perspective - published in 1927. Gleizes makes the crucial point that the reason for showing the full circle and not the perspective ellipse is not to convey more information about the nature of a cup but because the circle has more esemplastic (I feel free to use this invaluable word in my own voice) power than the ellipse. (48)

45   Miller: Einstein, Picasso, p.171.

46   René Huyghe: Histoire de l'art contemporain - la peinture, Alcan, Paris, 1935, quoted in Gray: Cubist Aesthetic Theories, p.74, though here the example taken is a table. Is it a trapezoid or a rectangle?

47   Severini: From Cubism to Classicism, pp. 73-5.

48   Gleizes: 'Peinture et perspective descriptive' in Puissances du Cubisme, pp.64-5.

Juan Gris' drawing, Man in a Top Hat of 1912, is a text book illustration of the device and the treatment of the top hat clearly resembles that of the hat in Metzinger's Portrait of Albert Gleizes, begun in 1911. (49) Gris' picture is ambiguous. It can be seen either as a joke - the habits of the Assiette au Beurre cartoonist dying hard - or as an unsuccessful attempt to do something he has not yet mastered. But if it is seen as satirical, he is satirising himself since the painting is done on the same principle as the unquestionably serious Portrait of Germaine Raynal or the Watch with Sherry Bottle. Fragments of the object are taken from different angles and inscribed in an easily readable grid of parallel lines. It is these lines that primarily determine the esemplastic power of the painting.

49   We might also note the hat of the Man in a Café and the preparatory sketch given in the illustrations to On "Cubism"

My suggestion then is that Metzinger studied with Princet in 1911 and that this contributed to the transition from the Nude of the Salon des Indépendants to Tea-time, shown later in the year in the Salon d'Automne. Gris, impressed by Tea-time, joined them and this contributed to his own development - parallel to that of Metzinger, though in the event Gris did not take it very far - of a grid based painting in 1912. 

One article by Princet has, as it happens, survived (50) - the introduction to the catalogue of a 1912 exhibition by Robert Delaunay and Marie Laurencin, artists we do not normally associate with interest in the new geometries though Delaunay did have a lively interest in scientific colour theory. It is a reflection on the relation between sensibility and intellectual rigour and, though it argues for intellectual rigour, it is by no means the sort of display of superfluous erudition one might have expected following the accounts of the old, Kahnweiler-based, school of Cubist historiography. It is quite clearly written from a love of painting and painting, not an intellectual theory, is the end it envisages. Princet's relations with Delaunay, and his reservations with regard to Picasso and 'the Cubists', will be discussed later in the present essay.

50  Hopefully a translation will be included in the present anthology.