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Art and Geometry: A Study in Space Intuitions

Art and Geometry: A Study in Space Intuitions

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by William M. Ivins

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One of Western civilization's jealously guarded myths is that of Greek cultural supremacy. In this controversial study, William Ivins shows that the limitations of the Greek worldview actually hampered the development of the arts and sciences and gives a stimulating history of the new ideas of the Renaissance, especially in painting and geometry, that freed us


One of Western civilization's jealously guarded myths is that of Greek cultural supremacy. In this controversial study, William Ivins shows that the limitations of the Greek worldview actually hampered the development of the arts and sciences and gives a stimulating history of the new ideas of the Renaissance, especially in painting and geometry, that freed us from ancient misconceptions.
Beginning with the Greeks, the author explains for the general reader the differences between ancient and Renaissance painting and sculpture, proving that the curiously static quality of Greek art arose from a misunderstanding of the laws of perspective. He then shows how this misunderstanding was corrected by Alberti, Pelerin, Durer, and other Renaissance artists who provided the first fruitful investigations of perspective. From there to projective geometry was but a step, and the author covers this major advance in our knowledge through the work of Nicholas of Cusa, Kepler, and Desargues.
This book is perhaps the only concise history in English of the development of mathematical perspective and projective geometry. But the author's ability to relate styles in art to advances in geometry and his ingenious theory of the modern "visual" worldview and the Greek "tactile" worldview mean that his book will be provocative not only to mathematical historians but also to art historians and to anyone concerned with the history of thought, from philosopher to layman.

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Courier Corporation
Publication date:
Dover Books on Art History Series
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Product dimensions:
5.50(w) x 8.50(h) x 0.28(d)

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Dover Publications, Inc.

Copyright © 1946 The President and Fellows of Harvard College
All rights reserved.
ISBN: 978-0-486-14358-3



My adventure began with some simple homemade experiments with my hands and eyes as the respective organs of the tactile-muscular and visual intuitions. Obvious and unexciting as their results may appear to be, their implications are of remarkable interest.

Unless the eye moves rather quickly, it is conscious of no breaks in the continuity of its awarenesses, although, of course, as seeing is a selective volitional activity, such breaks are many and great. Things emerge into full consciousness very gradually as they come into the field of vision. If, while looking straight ahead, we hold a hand out at full arm's length sideways behind the line of the two shoulders, we cannot see it nor are we visually aware of it. However, if we slowly swing the arm forward, we discover that long before we can see the hand we are visually aware of it. There are positions early in the swing of the arm from the side to the front in which we are not visually aware of the hand as long as it remains still but are aware of it as soon as it begins to wiggle or move. As the hand is swung forward from this range it slowly assumes shape and color, but the whole hand never attains its fullness of both for the unmoving eye. The smallness of the angle of sharp vision can be tested by looking fixedly at a short word in a line of text and noticing how quickly the type in the words to its right and left loses definition, and that there are no points at which this loss either begins or finally culminates in a complete failure of awareness. Because we can and do continually move our eyes without being aware that we are doing it, we are not usually conscious of this fading in and out, but we act on it continuously every time we walk down a crowded street.

If in a darkened room a red-point light is slowly swung from behind the shoulder line into the position of sharp vision for the unmoving forward-looking eye, it makes its coming into awareness in several different ways. It comes into awareness not suddenly but gradually, so gradually that it is difficult to say just when in its swing it does come into awareness. As it swings forward it gradually becomes more brilliant. Somewhere in the course of this swing a remarkable thing happens. When the eye is first aware of the dim luminous spot, the spot has a neutral greyish color. As it moves further forward it enters a region in which it changes color and becomes a full red.

To phenomena of this kind we must add the "after images." If we look fixedly at a pattern in bright red and then look at a piece of white paper we continue to be aware of the pattern, but as a pattern in green. If we look at a pattern of black squares separated by lanes of white, the lanes have flickering cores of grey. These things have much to do with the fact that our awareness of the hue of any particular portion of a surface depends in large measure upon its area and upon the areas and the hues of the neighboring portions and the comparative brilliance of their illumination.

In addition to things of these kinds there is the long series of visual effects that take place with changes in position. Objects get smaller and less brilliant as they get further away from us. Very distant objects are mere shapeless nubbins. Near objects continuously change their shapes as we move about them. A pine wood close to us is a mixture of deep greens and browns, but at a distance it becomes a diaphanous light blue. In the course of the day the whole landscape drastically changes its color. As the light decreases, the different colors disappear from view at different times. Parallel lines as they recede from us tend to come together.

Now, as against all this fading in and out, this shifting, varying, unbroken continuity of quite different visual effects, what do we discover when we examine the tactile-muscular sense returns given by the exploring hand? Doubtless these returns are extremely complex in themselves, but as compared with the visual returns they are definite, simple, and very restricted in gamut. To begin with, as we all know from our experiences in finding our ways about in completely dark rooms, tactile awareness for practical purposes is not accomplished by a gradual fading in and out of consciousness, but by catastrophic contacts and breaking of contacts. My hand either touches something or it does not. My hand tells me that something is light or heavy, hot or cold, smooth or rough. I can measure an object that is simple in form against a phalange of my thumb or a stick, and by counting my motions I can tell how many phalanges or sticks high or wide it is. Short of accident, my muscles tell me that these measurements always require the same number of movements, i.e., that the object does not change in size or shape. If the object is a molding, I can run my fingers or a stick along it and determine that within the reach of my hand its lines are always the same distance apart and do not come together, i.e., that the lines are parallel. Furthermore, the fact that I can touch an object, hold it, push it, pull it, gives me a sense that there is really something there, that I am not the sport of a trick or illusion, and that this something remains the same no matter what its heaviness or lightness, its hotness or coldness, its smoothness or roughness. Moreover, the shapes of objects as known by the hand do not change with shifts in position as do the shapes known by the eye.

The hand, however, as compared to the eye works only within the short limits of reachable and touchable form. When detail gets small, the finger tips are unable to read it. When a form is large the hand cannot read it unless it is very simple in shape, and if it is very large the hand can only read limited portions of it. The hand is unable to correlate a series of simultaneous movements of different elements, and can gain little or no idea of the simultaneity of change in the shapes of the muscles and the limbs and body of an animal in motion. If a form in decoration is very simple and is repeated, like the egg and dart or the wave fret, the hand can follow and recognize it, but if it is complex and constantly changing without repeat, as in a rinceau or on a Gothic capital, the hand is at a very great disadvantage. In any continuous pattern the hand needs simple and static forms and it likes repeated ones. It knows objects separately, one after another, and unlike the eye it has no way of getting a practically simultaneous view or acquaintance with a group of objects as a single awareness. Unlike the eye, the unaided hand is unable to discover whether three or more objects are on a line.

Because of the fact that we frequently see objects at the same time that we touch them, we are apt to associate two different groups of sensations, so that we say that something looks heavy or dry or cold, although the eye in fact is unable to know these sensations. But, it is important to notice that we never say anything feels red. Thus we are constantly giving visual expression to tactile qualities, but rarely or never reverse the process. The result of this is that, although the presence of a drawing on a sheet of paper escapes tactile detection, we can look at the drawing and say that it is composed in largest measure not of visual but of tactile awarenesses. Where an object visually overlaps another of the same hue and illumination its outlines are actually lost to sight, but our tactile intuition forces us to indicate its visually lost outline in our drawing.

As a result of all this I believe the tactile mind is very apt to be aware of and to think of things without any feeling of necessary relationship between them, except such perhaps as results from memory of an invariant time order of repeated but distinct and separate awarenesses.

Tactually, things exist in a series of heres in space, but where there are no things, space, even though "empty," continues to exist, because the exploring hand knows that it is in space even when it is in contact with nothing. The eye, contrariwise, can only see things, and where there are no things there is nothing, not even empty space, for that cannot be seen. There is no sense of contact in vision, but tactile awareness exists only as conscious contact. The hand, moving among the things it feels, is always literally "here," and while it has three dimensional coordinates it has no point of view and in consequence no vanishing point; the eye, having two dimensional coordinates, has a point of view and a vanishing point, and it sees "there," where it is not. The result is that visually things are not located in an independently existing space, but that space, rather, is a quality or relationship of things and has no existence without them.

From what has been said it is obvious that there repeatedly come crises when the visual and tactile-muscular returns are in conflict. When this happens it is necessary that we elect for one or the other as the test of "reality." As we habitually elect for one or the other so we make assumptions on which we base our philosophies and our accounts of the world. I believe that the ancient Greeks provide an unusually clear example of this.

As nearly as I can discover, the Greek idea of a primary substance, or matter, having extension and located in an independently existing space, but set off against a lot of mirage-like secondary, attributable, qualities, may be regarded as the reduction of the tactile-muscular intuitions to a sort of basic philosophical principle. In many ways it would seem that such a central idea is destructive of any sense of necessary unity or continuity between things. It seems to me that running through most of what the Greeks did I perceive qualities that are difficult to explain on any grounds other than that intuitionally the Greeks were tactile minded, and that whenever they were given the choice between a tactile or a visual way of thought they instinctively chose the tactile one. I can imagine nothing more antithetical to Greek thought than Nicod's remark that "the order of views thus becomes the only fundamental space of nature."

Greek philosophers used the analogies of imitative art to illustrate their teaching about the unreliability of sensuous, and particularly of visual, apprehension and knowledge. Visual art was imitative and as such was a falsehood, and Socrates-Plato even went so far as to take a moral stand against it on that basis. If I read the Republic, the Sophist, and the Timaeus, and Professor Cornford's commentary on the last of them, correctly, Greek painters and sculptors, and for that matter poets and dramatists too, were but imitations of Platonic ideas making imitations of imitations of Platonic ideas. I supose Plato, from his extreme realist point of view, would have called one of the Roman copies that today do duty in our museums for so much of Greek sculpture the imitation of an imitation of an imitation by the imitation of an imitation—the cube of an imitation multiplied by the square of an imitation, or imitation raised to its fifth power or dimension of falsity. I hesitate to think how many powers of falsity Plato would have needed for a so-called restoration of a Roman copy by a German professor. Whatever one may think of rigmaroles such as that which I have just perpetrated, they are more serious than they may seem, and they account very satisfactorily for the abstract lack of personality, the composite group-photograph quality, of much of Greek art.

In view of Plato's attitude towards art, little can be more interesting than the fact that over the door of his school there was an inscription which said "Let no one destitute of geometry enter my doors." In the Republic, 527, he says of geometry that it is "pursued for the sake of the knowledge of what eternally exists, and not of what comes for a moment into existence, and then perishes," and that it "must tend to draw the soul towards truth, and to give the finishing touch to the philosophic spirit."

The contrast drawn by Plato between the falsity of art and the truth of geometry was seemingly a commonplace, and so long as that dichotomy existed there was little chance that the similarities between the two would be recognized. Today, however, the situation has vastly changed. In the eighteenth century Berkeley made his destructive criticism of the distinction between the primary and secondary qualities or characteristics. In the nineteenth century and in this one, study of the relationships between logic and mathematics has brought about a much deeper understanding of what mathematics is. Today geometry has ceased to be The Truth and become a form of art marked by lack of contradiction above rather a superficial level. Thus, G. H. Hardy refers to "the real mathematics, which must be justified as art if it can be justified at all." We know that the number of possible geometries is very large, and that any of them as a form of mathematics is only "true in virtue of its form." Just what "true in virtue of its form" may mean seems not to be definitely known, but short of some very unforeseen implications it is probable that it applies equally well to both geometry and representational art.



If we want to think clearly about art we must adopt as a guiding principle some sort of what the logicians call a doctrine of types. Otherwise we are all too apt to find ourselves talking solemnly, and even heatedly, about square circles and pink voids.

In the presence of various types of art based on different assumptions—e.g., Egyptian, Greek, Medieval, etc.—we can distinguish between them, discover their respective qualities, and explore their relations to ideas of all kinds, and we can, possibly, as a result of the study of their bases and their structures, say that one type of art is more primitive, less complex, or in various ways more restricted than another. To say, however, that an art of one type is intrinsically truer, better, finer, more beautiful, or more important than an art of another type, or that it would be improved by the addition of a missing quality or the deletion of one that is present, is to make statements that are meaningless. Moreover, when, in making great claims for an art, we deliberately leave out of our accounts moral and intellectual factors that are uncomfortable to discuss, we deliberately falsify.

It is always necessary to remember that a people's actual valuations of the past arts of its own tradition are to be sought not in the appreciative words it utters but in the art that it currently produces. In this respect art is like logic, the history of which is to be sought in what men have done and not in the books they have written on logic. Every broad change in a living art is accompanied inevitably by a revaluation of the arts of the past, which frequently is of the most drastic kind. Such changes and revaluations are really the discoveries of new values, and, without them, art as vivid expression, far from having a history, could not exist. In the intervals that occasionally intervene between the exhaustion of a set of values and the discovery of a new one, what is called art is merely a skillful but dull and lifeless professional practice. The germs of the new values are usually to be found in what most people at the time regard as fumbling and incompetent performance, and what many people in later years may regard as decadence. Their actual character is not recognized for a long time to come.

In view of what I have just said, I shall not discuss or compare the aesthetic merits or demerits of either the Greek or the modern accomplishments. For the time being my interest is simply to discover and bring out, as best I can, the differences between the basic sensuous intuitions of the Greeks and of the moderns as exemplified in their so very different arts and geometries. The most direct way of doing this is by looking for qualities that are markedly present in one group and missing in the other. I shall, therefore, as a modern, content myself with pointing out the absence of certain significant qualities from both the art and the geometry of the Greeks, and then give a hasty sketch of how those qualities emerged in later times.

I shall not attempt to define the Greeks, or say where, when, how or how wonderful, etc., etc., they were, but shall take them frankly, in one of their own phrases, as an "undefined common notion." However, when we try to think about any art it is always well to know, if we can, something of the history and the animus of the accepted ideas about it. I shall, therefore, point out briefly several things that are profitable to have in mind when thinking about what we think about Greek art.


Excerpted from ART & GEOMETRY by WILLIAM M. IVINS JR.. Copyright © 1946 The President and Fellows of Harvard College. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Art and Geometry: A Study in Space Intuitions 4 out of 5 based on 0 ratings. 1 reviews.
A_Sloan More than 1 year ago
Dismantling the Grecian Impact on Western Culture The study of geometry in respect to art has never been as stimulating as in William Ivins' book, Art and Geometry: A Study in Space Intuitions. Readers will be surprised to see Ivins debunk commonly-held notions about art and math from Grecian culture as he begins by explaining "simple homemade experiments" to more complex research. He delves into Artistotle, Plato, the "falsity of art" and the "truth of geometry" as each component relates to another. One of the most surprising components of Ivins' book is his ability to break down classical Greek ideas regarding composition in a way that makes sense for the layman. This book takes a fresh approach toward topics like Greek philosophy, logic, and science, tearing apart many components of art and math that others are afraid to each approach. Even those who do not agree with his message will find it to be a remarkable and intriguing read. While this book might be short, it is packed with information about Grecian art, math, and culture that still impact the thriving nature of Western culture. Readers will end the book feeling that they have a better grasp of the natural world as it actually exists, not as it would be easier to manager. A great follow-up in my opinion would be The Painter's Secret Geometry, as it will utilize the understanding as a lens to then explores paintings by the masters.