Martin Kemp: Linear Perspective from Durer to Galileo

(this is Chapter Two of Martin Kemp's "The Science of Art".
Quoted text is in YELLOW.
Text quoted from other authors is in GREEN)

Durer (1471-1528): Presentation in the Temple, 1504


I hope to indicate that there are certain patterns to be discerned in those countries in the forefront of the quest for what we may call 'scientific illusion' in art.

Kemp must have been reading the Cataline speeches of Cicero. Whatever the orator says often enough, eventually the audience will suspect that it may be true.



The Presentation in the Temple shows how perspectival devices can be utilized powerfully in a narrative context -- to convey a sense of the solemn and imposing dignity of the ancient temple


But then, Kemp notes that the coffering in the ceiling is incorrectly drawn, and shows a "corrected version" done by Jean Pelerin only 5 years later.

Pelerin assumes that the ceiling bays are all squares of the same size, which would result in the rear bay appearing much smaller, and would move its supporting column much further to the right, substantially changing its relationship to the figures in the foreground.

Kemp allows that Pelerin's "correction" might be considered pedantic, and even detrimental to the design, but then quotes this passage that Durer wrote in a letter in 1506:

I shall be finished here in ten more days, then I shall ride to Bologna where someone is willing to teach me the secrets of perspective. I intend to stay there about eight days. Here I am a gentleman, at home I am a parasite."

And then the speculation begins.

Who was his teacher - and what was he taught?

But as Durer's concern about being perceived as a gentleman might suggest, these lessons in perpective may have been more connected to his interest in social status than in enhancing his paintings - much as students in MFA programs today learn more about how to talk like an artist rather than about making things. Durer needed to fit into the humanist princely courts of his day - just as our conteporary artists need to fit into university art programs.

Or, maybe he was just curious about mathematics.

In 1525 he published his own treatise: "Instuction in Measurement with compass and ruler in lines, planes, and solid bodies", directed at young German artists who "have grown up in ignorance, like unpruned trees. Although some of them have achieved a skilful hand through continual practice, their works are made intuitively and solely according to their own taste"

The work which can most obviously sustain a full-scale analysis is his brilliant St. Gerome engraving of 1514. This is his supreme demonstration of the engraver’s arts in union with the perspectivist’s scientia. Using a radically off-centre viewpoint he has openly delighted in extremes of transformation and plunging space, providing a strenuous course of perspectival gymnastics for the observer’s eye.

To me, this appears to be a rather straighforward application of an Albertian picture box, except, of course, that there is no pattern of square tiles on the floor or ceiling.

Kemp believes that he has found the remnant of one, in that the opposing corners of the table create a diagonal line that intersects the diagonal connecting the bottoms of two chair legs, and that intersection is on the same horizon line established by the vanishing point from all the receding orthogonals of the wall and ceiling. But even if that is not a coincidence, just how, exactly does that enhance the sense of space in the room? It certainly does not incrementally measure it out, the way that a grid on the floor or ceiling would.

He also notes that the spatial construction is not accompanied by a geometric pattern on the surface of the picture plane, as he finds in the paintings of Italian masters like Piero. And he feels that the orthogonals are "plunging in a willfully independent way to their common vanishing point" , the features in perspective seem "more obtrusive"and sometimes "overtly peculiar" with a feeling of "dynamic freedom".

So he wants to talk about the emotional effects of composition.

But how can this topic be raised without reference to narrative -- i.e. what is St. Jerome doing in that pleasant, sunlit room, with a lion and mangy little dog on the floor?

With neither the narrative features (that Smith would have discussed) or the compositional problems (that interested White), Kemp's discussion of Durer and his St. Jerome was a bit thin, and I'm not sure why he discussed him at all, other than his status as
"the last major painter who was also a significant geometer"

But I am very grateful for his introducing me to another Nuremburg artist, the metal worker, Wenzel Jamnitzer, who published a book of delightfully whimsical engravings, the Perspectiva Corporum Regularium in 1568.

Apparently, these geometric solids represented the various Platonic states of matter.

But more than that, they are just fun to look at, and remind us that, for whatever reason, Germans make such good machinists.

Here is what Wenzel did for a living.

Jean Pelerin (1445-1524)

Here are some illustrations from yet another book about perspective from the early 16th C. - this is the fellow who "corrected" the Durer woodcut shown earlier.

Some very simple, enjoyable views, that would be commendable if done today, though in his era of great book illumination, it's quite modest.

Jean Cousin (1500 - 1593)


And here's another guy who wrote a book about perspective

In this case, he was also a great painter, but unfortunately, there aren't any perspective views of buildings in his paintings so Kemp has no reason to discuss them.

The next painting he discusses is Raphael's "Marriage of the Virgin" which demonstrates "an impressive mastery of the orthodox description of architectural forms in space" and "a sense of mathematical harmony worthy of his great predecessor"

Kemp offers no further discussion for this piece, but I can't help but digress into comparing it with its prototype done by Raphael's teacher, Perugino. four years earlier.

Raphael has given that gazebo in the background 8 sides instead of six, which does, I suppose, make its rendering in perspective a little more complicated. He has also set it further back, giving more space to the intervening plaza. But that extra space does present a problem for the Albertian system that uses only one vanishing point. Those extended horizontal lines in the pavement badly need to curve back,especially since they are set against the curving edges of the octagon behind them. But Raphael has kept them straight, making the scene feel unreal.

But on the other hand, Perugino has certainly made his version feel unreal as well - with the priest appearing to be some kind of magician, the center of this little drama, while Raphael has given greater emphasis to the Virgin.

Why has this wedding ceremony been set into a palatial garden that includes a public square and monumental gazebo? That's a question that Norris Kelly Smith would have liked to answer.

Kemp next brings our attention to the faux architecture that Raphael painted in the Loggie of Leo X at the Vatican. He claims that it applied a "new mode of operation", but does not explain it.

I couldn't find any images on the internet, except for one of the scenes, shown above, that was set into the architectural framework of the ceiling.

Carlo Urbino

Further discussion then follows regarding that golden age of illusionistic ceilings of the 16th C. Were they designed using projective geometry -- or by building a miniature architectural model and viewing it on an angled mirror ?

Carlo Urbino is credited as being the author of a practical handbook for how to make them -- but I'm only interested in these illusions if I find the painting as a whole to be compelling, and I think there's good reason why Carlo is known only to specialists.

I'd like to read some further speculation on why Italians of that era were so fond of illusionistic holes in the ceilings of their churches - but Kemp doesn't go there.

As Kemp notes, the fondness for illusion in that era led the humanist-aristocrat Daniele Barbaro to commission Veronese's decoration of his villa in Maser, so wonderfully celebrated 400 years later by Josef Losey's film of Don Giovanni (where you might note in the linked video, Losey uses actors to create the illusion of an illusionistic painting)

But as Kemp also notes, Barbaro seemed to be more interested in geometry (he wrote and critiqued essays on this subject) than in aesthetic effect, aand he scolded artists for being "content with a simple procedure which misses the true divinity of geometry". Kemp then looks at the writings of Egnatio Danti. His brother Vincenzo, was a great sculptor, but Egnatio was a priest who served as a university professor of mathematics as well as a consultant to the Pope. So he is far less interesting to me than to Kemp who calls him “the most intelligent, useful, and thoroughly informative book on perspective construction to have appeared at that date”

At least his next author, Giovanni Paolo Lomazzo (1538-1592), was a painter until he went blind, but even then Kemp does not discuss his paintings. Lomazzo wrote "Trattato dell'arte della pittura, scoltura et architettura" which Wikipedia calls "a milestone" in the development of art criticism, but apparently it has not been considered important enough to have been published in an English translation. According to Kemp, Lomazzo had some difficulty with Michelangelo's dictum that the artist should have calipers in his eyes rather than base his proportions on numerical or geometric calculations. How could that super-human artist be wrong? Apparently Lomazzo suggested that Michelangelo's dictum only applied to those who were as deeply schooled in the rules of perspective as he was. BTW, Lomazzo also believed that the eye emits rays and vision is the result of those rays bouncing back to it - a theory which was restated a few years ago by an Indian fellow on the Aesthetic-L discussion list.

Federico Zuccari (1540-1609), the next author Kemp discusses, was also an artist, and even if he wasn't as good as Michelangelo, at least he agreed with him:

I say strongly, and I know I say the truth, that the art of painting does not derive its principles from the mathematical sciences and has no need of recourse to them to learn the rules and means for its practice, for art is not the daughter of mathematics, but of nature and design

I happen to agree with Zuccari, and obviously if Kemp did as well, the book I am now discussing would not have been written.

And BTW, a similar assertion has been made regarding the practice of scientific research: No practicing scientist learned the craft of research from books or articles, Polanyi argued. Rather, they had to practice craftlike skills, which they internalized via social relationships like apprenticeship training. Scientists developed an aesthetic sense for what counted as good science, according to Polanyi, and used any means available to convince colleagues from rival research schools to believe a given result. Scientists often formed their beliefs from an immersion in particulars that resisted explicit articulation; he likened the experience to religious conversion. To Polanyi, the routines of scientific research could never be captured by recipes, and therefore any effort to steer the direction of research, or subject science to central planning, was bound to fail. As Kemp notes, subsequent treatises on projective geometry tended to ignore pictorial applications and moved on to issues of ballistics and astronomy.

But the very first art academies, like the Accademia Del Designo in Florence and the Accademia di San Luca in Rome did indeed offer instruction in mathematics.

And as Kemp has us note in the above detail from a painting by Lodovico Cigoli, the Virgin is standing upon a moon that is as pockmarked as Galileo described it in his calculations from the shadows observed on its round surface.

Observation and calculation as practiced in the beginning decades of the Scientific Revolution did indeed have at least some impact upon artistic practice. Though it seems rather minimal to me - since, after all, no telescope has ever seen the Virgin standing in heaven.

As it turns out, Cigoli and Galileo had quite an ongoing correspondence, as they both shared an interest in mathematics and representation. Galileo was even elected to become a member of the Accademia del Designo in Florence, though that did not prevent him from writing the following two decades later: "Philosophy [i.e., physics] is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth."

Which seems, at least to me, to be as direct an attack upon the mimetic visual arts as could possibly be made.

Here's a drawing by Cigoli, showing both his strength in figurative detail and his weakness in spatial design.

He was almost an exact contemporary of Caravaggio who was born 10 years later but died before him.

So ends Kemp's discussion of "Linear perspective from Durer to Galileo" - a title which, one may sadly note, ends with the name of a non-artist, reminding us that Kemp is not focusing on visual art, as Smith and White had done.

He also does not seem to be doing a thorough, insightful history of geometry, but I admit that this subject does not interest me enough to re-create the various methods illustrated by the diagrams that he has included.

Martin Kemp: Linear Perspective from Brunelleschi to Leonardo

(this is Chapter One of Martin Kemp's "The Science of Art".
Quoted text is in YELLOW.
Text quoted from other authors is in GREEN)

Giotto: Confirmation of the Rule, Bardi Chapel

Before launching into Brunelleschi, Kemp takes us to his predecessors (with a footnote acknowledgment of John White), but only to note the use of rules --- as laid out in Il Libro Dell'Arte by Cennino Cennini (1370-1440)



And put in the buildings by this uniform system: that the moldings which you make at the top of the building should slant downward from the edge next to the roof; the molding in the middle of the building, halfway up the face, must be quite level and even; the molding at the base of the building underneath must slant upward, in the opposite sense to the upper molding, which slants downward. Chapter LXXXVII

.... and then as Kemp has restated, and expanded, them :



“those lines and planes situated above eye-level should appear to incline downwards as they move away from the spectator, those below eye-level should incline upwards, those to the left should incline towards the right; those to the right should incline towards the left; there should be some sense of the horizontal division and the vertical division which mark the boundaries between the zones, and along those divisons the lines should be inclined little if at all.


Regarding the above example, Kemp has noted that:


The convergence is not perfect but it looks too organized to be the result of chance or even ‘judgment by eye’.


Is 'judgment by eye' really such an ambiguous term that it deserves those scare quotes ?

If it wasn't judgment by eye, what system was used? Might there have been a conscious decision to have most of the receding othogonals converge on a single point, except for a few?

Giotto (or follower), Christ Among the Doctors, Assisi


In this example, as Kemp shows us in a diagram, all of the othogonals converge precisely at a single point above Christ's head, while the lines drawn through the diagonals of the coffered ceiling converge at two distance points, equally spaced to the right and left of center.


There are no tiles on the floor to measure out the bottom of picture box, but wouldn't this still serve as an example of "linear perspective" as Kemp has defined it, about a hundred years before Brunelleschi?

Kemp has a problem with it as follows:



Although more developed geometrically.. the space as a whole is much less coordinated and the motif on the ceiling gives the effect of a technical device rather divorced from the figures below.

But are not "divorce" and lack of "coordination", as Kemp sees it, aesthetic issues that have nothing to do with the proper practice of linear perspective?

It looks pretty weird to me, too.

Judging from the reproduction, the heads of all the fellows against the wall feel stranglely enlongated, as if they were seen through a fun-house mirror.

The Virgin has a very large head, and the size of the figures do not seem to diminish in proportion to distance.

None of which would necessarily be an aesthetic disaster -- but it does seem to me that one specialist painted the architecture, and then another fellow came in to do the figures in front of it.

But again -- this is not an issue of linear perspective as Kemp has defined it.


Also digressing from his topic, Kemp offers this explanation for the far-right POV in the Bardi St. Francis fresco shown above:



This sense of the eyewitness character of Giotto’s scene reflects one of the major motives behind the new naturalism. This motive was the desire, in a particularly Franciscan spirit, to present the sacred narratives to the spectator on human terms..”


It does seem that Renaissance pictorial space owes more to Franciscans than to secular humanists, or a scientific curiousity about optics.


Duccio, Temptation of Christ on the temple




Duccio similarly achieved remarkable passages of spatial description, observing the general rules of convergence to achieve schemes which only barely fail to obtain geometric precision and display a responsiveness to subject, scale, and location in no way inferior to Giotto’s. The particular problems tackled by Duccio in portraying the octagonal building in his Temptation will become particularly relevant when we come to look at Brunelleschi’s invention”

John White discussed this painting here , and as I noted, the POV for the octagon is far right of the POV for the octagon as a whole, probably for narrative reasons. (i.e., in the struggle between Christ and the Devil, the viewer wants to be on the side of Jesus)

Kemp adds that these artists were more concerned with making
"effective, functioning, devotional pictures than with clever passasges with isolated and perhaps even disruptive design" - but then, why should we be concerned with whether they fail to obtain geometric precision ?


Pietro Lorenzetti, Birth of the Virgin

Both John White and Norris Smith discussed this painting a some length here and here

Kemp notes that although "The underdrawing .. testifies to the immense effort of constructive geometry which has gone into the perspective effects" ,and "he has reasoned that the higher plane of the bed cover is seen at a flatter angle, and has accelerated its perspective accordingly, nevertheless, this was done "incorrectly by strictly perspectival principles" and the off-centered rib vaults were "created on an intuitive rather than calculated basis."

Leading Kemp to conclude that "generally most artists seem not to have been attracted by a method which promised much labour for an optical reward which was as yet of uncertain value."

and
We can in retrospect see what remained to be accomplished, namely the demonstration of an internally consistent system for all the spatial elements in a picture and above all, a proof that the system rested upon non-arbitrary foundations.

Kemp likes to hedge his bets.

On the one hand, he acknowledges the sophistication in accomplishing narrative purposes of these paintings -- but on the other, they do not exhibit an internally consistent geometric system, which provides the only kind of foundation which Kemp would call "non-arbitrary"

And on the one hand, he credits Brunelleschi with the invention of just such a "non-arbitrary" system, but on the other he acknowledges that the only evidence of this achievement is found in a text written by Manetti who "gives no indication of how Brunelleschi achieved his results".




And, in concert with Smith's assertion, "Brunelleschi’s measured representation of these two revered buildings was deeply locked into the system of polical, religious and intellectual values shared by those who exercised the greatest influence on Florentine civic life in this period”




Donatello, St. George

(White's discussion is found here )

Kemp notes that in this relief, Donatello
"created an unprecedented sense of atmospheric space", though he also observes that “there is nothing, in terms of measured perspective that goes beyond trecento methods – but an ambitious mind is obviously at work”

So I'm wondering why he mentioned this piece at all.


Donatello, Feast of Herod



"The focus of this upper part is, however, conspicuously above and marginally to the right of the oint to which the tile system converges. The two spaces are not, therefore, perspectivally united – was Donatello unaware, as his trecento predecessors had been, that all the features running into the space at right angles to the picture place should converge to the same point in perspectival projection ?"

This presents a real problem for Kemp, and he won't let Donatello off the hook for "artistic license" or "tampering with rules before we can be sure that he understood them"

Kemp sees this piece as "genuinely unresolved"

Would he suggest, therefore, that the piece would be better if Donatello could have managed to have all the othogonals converge at a single point?

That's quite a bold assertion.

Massacio, Pisa altarpiece


Masaccio, Trinity

(John White discusses Masaccio here )

I would like to point out that a consistent, one-point perspective system was applied at Assisi about a century earlier.

And, just as with that example, Masaccio does not offer us a floor grid.

So, what exactly is his contribution to this technique ?




When we look at a painting such as Masaccio’s we make an automatic and required assumption that we are seeing regular objects under spatial distortion.

With absolutely no evidence outside the painting to justify it, this assumption is rather difficult to defend, so Kemp has aggressively stated that we are required to make it.



The reader should be warned that Masaccio’s construction is not entirely consistent and regular in all its details and no reconstructive strategy can eliminate all the problems.

Kemp even acknowledges that if Massacio really were trying to present "regular objects under spatial distortion", he has failed, since nobody can reconstruct just what and where those regular objects were supposed to be.

Then, like many art historians before him, Kemp launches into his own speculations on what went where.

There is no mention of a sacred narrative until he reaches his conclusion as follows:



Masaccio’s differentiation in depth between the all-too-mortal world this side of the plane and the indubitably spiritual realm inhabited by God, Christ, Mary and John is an act of high genius. A new spatial technique has simultaneously been mastered technically and locked into the meaning o the picture in a way which only a few artists have since equaled

But does that effect require the viewer to be aware of exactly where all those objects would be located in a floor plan of the imaginary chapel ?


***************************


At this point, Kemp introduces Alberti and details the geometric construction of his grid.

(John White's discussion of Ghiberti is found here )

Kemp introduces Ghiberti as a "Renaissance intellectual" rather than a "late-Gothic craftsman", and notes the haphazard geometry of the ceiling tiles in the "St. John before Herod" as compared with the Albertian geometry of the floor tiles in "Jacob and Esau" that was done eight years later.

And, indeed, Ghiberti does seem to fit Kemp's model of the artist-scientist,as he compiled the extant literature on optics, such as it was, and wrote the following in his commentaries:


“In order to always understand first principles, I have striven to investigate in what manner nature functions in itself, and how I may be able to apprise nature, how the incorporeal images of objects come to the eye, and how the visual power operates, and how the visual sensations arise, and by what means the theory of the arts of sculpture and painting may be formulated”



But did he actually add anything to mathematics or any of the natural sciences - or was he striking an attitude that was fashionable in the Italian courts of his time, appropriate for a man who aspired to a higher social standing ?

Piero Della Francesca, Flagellation, 1460

Piero was another artist who wrote about perspective, and Kemp concludes that "the flavour of all this is clear: the deductive perfections of mathematics provide an active, a priori model for our understanding of experience rather than arising simply from an empirical study of the sensory world”

Piero, himself, wrote: “many painters dismiss perspective because they do not understand the power of lines and angles and how they are obtained”

Piero's book demonstrates the use of geometry to plot out points in the human head, costume, and architecture --- but how much was it used in the contruction of this painting?

It seems to be a kind of "how to" book, like Cennini's, except that it specializes in the spatial geometry of specific kinds of objects, especially architectural features like vaulting or octagonal buildings.

It's what engineering students learn how to do in drawing class. I remember doing both isometric and perspective projections in one such class about 40 years ago.


And we can well believe that if had ever presented himself as a scientist, the way Ghiberti did, Kemp would have let us know about it.

To me, the "Flagellation" has that cool, rational feeling of computer generated images.

But it's far more beautiful, reflecting a measuring of tones, colors, and proportion that software cannot perform.

And there seems to be no concensus concerning the identity of any character in the scene other than Christ.

Several interpretations suggest that the painting compares the sufferings of Christ with the barefoot young man with a laurel halo in the foreground, and that makes sense to me -- as a kind of tribute to whoever, or whatever, that young man may be.

It's too bad neither Smith nor White discussed Piero. He seems to have been more concerned with composing volumes in space than in presenting dramatic action figures. And his space is so distinctly delicious.
I'm sure Smith would have reminded us to think about the classical, palatial architecture as the proper setting for whatever public discourse is here being presented.

Kemp offers the above "Resurrection" as an example of "anatomical stereometry" - especially in the soldier's head which is leaning back against Christ's staff. It does indeed closely resemble one of the studies shown in Piero's De Prospectiva Pingendi.

But one might notice the total absence of any receding orthogonals, and the consequent shallow space.

Kemp concludes that: "The expression of sensual pleasure through ice-cold logic provides the dominant effect of his art." , which also seems to describe the aesthetic of classical statuary set into a Greek temple.

Domenico Veneziano (b. 1405-1410), St. Lucy Altarpiece (1445-47)

Kemp skips over Masolino, Fra Angelico, Filippo Lippi (all of whom White discussed) because they "did not add significantly to the scientific or technical means of perspective itself"

As we might recall, White discussed how these painters reconsiled pictorial space with the integrity of the surface, an aesthetic concern of the 1950's which Kemp does not address.

But Kemp does focus on Veneziano's St. Lucy alarpiece.
(Smith discusses it here )

"Not only does contains a virtuoso display of advanced perspective, but also exploits a marvelously cunning series of visual conjunctions which compress elements at different depths into an interlocked composition"

As Kemp explains it, the "virtuoso display" is the geometry required to construct the receding lines of that complex pattern of tiles on the floor, while the "interlocked composition" is caused by those three arches whose tops appear to be at the surface of the picture plane, while their bottoms are set back further into pictorial space.

Does that really "add significantly to the scientific or technical means of perspective itself" ?

Uccello, Nativity

Uccello, the Deluge

Uccello, Desecration of the Host

Other artists were more specifically concerned with the problems and conundrums posed by the Brunelleschian and Albertian systems. No one was more sharply alert to such conundrums than Paolo Uccello

The problem with the first example Kemp shows, the Nativity, is that hardly any of the paint is left. We can still make out the frame of the creche and the incised lines on the floor that recede to a different vanishing point. But all sense of pictorial space has been washed away. Kemp speculates that Uccello has invented a double P.O.V. system.

The problem with both of the scenes from the Desecration of the Host series is that the diagonals of the floor tiles do not recede in straight lines. Was Uccello unaware of Alberti's technique for drawing them ? Did he choose to ignore it? Was he inventing something new? Who knows.

But I do think that if the tiles were strictly arranged as an Albertian grid, the small paintings would lose some of their cranky, creepy livliness.

And then we have the enigmatic "Deluge", with separate vanishing points for the left and right walls and various indivisual objects drawn as perspective projections. Kemp concludes that
The whole exercise exudes an air of manic desperation as Uccello has attempted to weave the diverse motifs into some kind of coherence

Which would serve quite well for the desperate action being presented.

Was Uccello working out conundrums in Albertian perspective -- or was he just applying that system piecemeal wherever it might enhance the narrative or design ?

Donatello, Assumption of St. John, 1430


Next, Kemp revisits Donatello with a rather egregious display of double talk.

There is hardly a relief which does not deserve sustained analysis for the way in which space is charged with meaning.

And yet, he does not delve into meaning at all in his analysis of the above piece, the only one he has chosen here to study. He merely details its features.

He tells us:
This is not to say that he neglected to give deep consideration to the operation of the eye in the world. - but he never explains what evidence he has for such a statement.

He concedes that Almost none of this is conducted according to specific rules.
-- but if that's the case, why has he chosen to discuss Donatello instead of all the other 15th C. artists whom he ignores because they had nothing to add to the science of optics or perspective ?

And he even repeats the story told by Pomponius Gauricus who quoted Donatello as saying: “I have no other calculator than that which I always carry without fail amongst my belongings. Now if you wish to see it demonstrated, young man, bring me paper and pen. Donatello proceeded to draw figures nude and clothed.

Then throws in a quote from Michelangelo,
“the true artist must possess compasses in his eyes”

(a wonderful quote, though he does not share its source. I think I'll post it in the sculpture studio at the Palette and Chisel , to encourage students to leave their calipers at home)

Kemp responds to these quotes as follows:
This formula does not mean.. that unaided and licentious judgment were the order of the day, but that the science of art should be so assimilated by the artist that it could be put to use in whatever manner was appropriate for each instance”

Effectively removing any way to distinguish a practice based on science from one based on experience and intuition, and undermining, yet again, the premise of this book.

But even if great artists of the time disdained it, Kemp details how Pomponius Gauricus, and other humanists of his time, were indeed fascinated by the geometry of perspective and demonstrated some of its methods.

Which shows that the humanism has not changed much over the past 500 years as its scholars moved from princely courts to universities.

In as far as the means used by Mantegna were in principle already established, neither nor his followers will be major focuses of our attention, but in that he realized new applications we certainly should not ignore him......For the first time in a surviving scheme, perspective is used in a fully illusionistic manner to dissolve the plane of a vaulted ceiling

Kemp then speculates on what kind of reality might have resulted in this view-- and finds some problems with the size of the ledge on which the putti are standing (it's too small)

He also speculates on how Mantegna managed to foreshorten his figures.

"too good to have been left to chance"

But has he already forgotten what Donatello and Michelangelo told him?

An good artist's sense of proportion is hardly a matter of chance - but neither must it depend on measuring tools.

Kemp even acknowledges that
"Mantegna does not follow through on the full iimplications of the foreshorening" - because, of course, it would not have made the ceiling look any better.

Finally, we get to Leonardo, who, as a polymath in the arts and sciences, is really the primary focus of the topic of this book.



"Perspective is a rational demonstration by which experience confirms that the images of all things are transmitted to eye the by pyramidal lines. Those bodies of equal size will make greater or less angles in their pyramids acording to the different distances between the one and the other. By a pyramid of lines I mean those which depart from the superficial edges of bodies and converge over a distance to be drawn together in a single point. A point is said to be that which cannot be divided into any part, and such point, located in the eye, receives all the points in the pyramid"


Kemp does not give a date for the above Leonardo quote, though indicates that it came from his earliest writings.

Was it just a note to himself, or did he intend to share it with other scholars interested in the same topic?

As it casually conflates geometric definitions with the biology of the eye, it does not seem to be the beginning of a serious, scientific study. He doesn't really believe that the eye contains a point which cannot be divided into any smaller parts, does he? How could he prove it?

The Annunciation was painted when he was 20 years old, and "was designed in conformity with what a young artist was expected to know about pictorial space"

But those tiles in the floor do have some anomalies - like the "wild line" (as Kemp calls it) of one of the diagonals that goes to its own distance point - and the apparent curvature of some of the lines.









The height of his involvement with the painter's pyramid .... culminated in the Last Supper

It certainly appears to me to qualify as an Albertian picture box, and tto Kemp it "appears on first sight to be an uncomplicated expression of the rationality of the painter's science in the context of natural law"...... but ..... "closer analysis reveals a series of ambiguities and artifices which save the appearence of optical legitimacy while acknowledging the inbuilt problems and contradictions of perspectival illusion in a given situation"

The problem is the ceiling coffers.

Kemp goes though the various configurations of reality which they might suggest, and each variant has its own problems.

But then he suggests "the problems presented by the Last Supper so analysed might lead us to wonder whether we are asking a question which the painting was not designed to answer" -- and that is certainly the conclusion that I would draw.

It does seem to be a bit sacreligious to be puzzled by the possible dimensions of a fictional room when it contains the final meeting of Christ and the apostles

"Leonardo has used perspective for pictorial suggestion rather than absolute definition' , and I can't think of a single painting or drawing that was not - other, of course, than those made for technical purposes by engineers or architects.

Kemp implies that Leonardo began with the absolute and then "subverted" it,
and "admitted ultimate defeat" in order to achieve a more successful illusion.

But the only evidence he brings forward for that sequence is the distinction that Leonardo made in his notebooks between "artificial perspective" and the "natural perspective" which recognizes that the angle of the visual pyramid grows smaller with distance in all directions, not just the distance behind the picture plane.

Kemp began his discussion of Leonardo by listing the various topics in optics, geometry, and artistic practice that Leonardo pursued (none of which, by the way, concerned faith,religion, or sacred narrative) and aserted that he considered them "integral components in a great continuum of causes and effects"

So presumably, if hadn't thought about geometry and optics, his paintings would have been different.

But this is nothing more than an article of faith.



(in this chapter Norris Kelly Smith tells us that "Leonardo made less use of "mathematical" perspective than did any major artist excepting Michelangelo" - though he does not examine this painting in detail)

Kemp concludes that Leonardo was the one who
"most completely realised and investigated the multifaceted nature of Brunelleschi's invention -- he did not so much add science to art, or even art to science, as show how the 'science of art' possessed a creative unity of its own special kind in relation to both form and content"

But he has not shown how that "own special kind" is remarkably different from the other 15th C. painters whom he has mentioned.

He claims that all these other painters
"had, to their own satisfaction at least, isolated what they considered to be the salient factors in the analysis and construction of space -- by emphasizing the mathematics of the intersection largely at the expense of the physiology of vision"

But Kemp's narrative is entirely fictional, with no corroborating evidence being offered.