Vermeer

Vermeer and the Art of the Renaissance Perpective - Part 1. (UPDATED)

Introduction

Relatively little was known about Vermeer's life until recently. Seventeenth-century Dutch master Jan Vermeer (1632-1675) seems to had been devoted exclusively to his art and his family. He was recognized during his lifetime in Delft and The Hague, but his local acclaim gave way to oblivion after his death. Today along with Rembrandt, he is considered the greatest Dutch master and it is not hard to understand why. Although, Vermeer does not manifest as much passion and use of dramatic light as Rembrandt, the emotional effect of his paintings is deep and lasting.
Vermeer's strokes are smooth and precise, and his method is slow and tedious. He painted relatively few pictures, mostly a typical Dutch room, where one or two characters deal with a common household activity. But the longer we look at his paintings, the more they draw us in and the more we realize that these beautiful lifelike scenes represent a real challenge.

It is generally considered that Vermeer used a camera obscura in the process of creating a realistically depicted space. Camera obscura, the grandfather of the modern camera, was popular in the seventeenth century in Europe. It is likely that Vermeer knew this device and in some of his paintings he may imitate the visual effects produced by the camera obscura, but the space constructed in his artwork is purely geometric.

The Development of Delft's Perspective

The invention of a linear perspective in early quattrocento became an indispensable skill for every ambitious artist. In Delft, at the beginning of the second half of the 17th century, artistic feelings had begun to change and a new style was born. The painting of religious subjects declined and a new market for secular subjects sprang up. The stylistic conventions were replaced by new approaches. The architectural painting, portraiture and, above all, the domestic genre was gaining in popularity. New young artists moved to the city and were changing the stylistic direction.

The great influence on Vermeer's perspective perception was the renowned Delft artists who had devoted themselves to painting in which knowledge of perspective was indispensable. Gerard Houckgeest and (1600 - 1661) and Emanuel de Witte (1617 - 1692) and Hendrick Corneliszoon van Vliet (1612 - 1675)expanded on possibilities of architectural painting and their depiction of church interiors pulsated with accuracy of a real space. The most prominent representative of this genre, Pieter Saenradam (1597 - 1665), visited Delft in 1651 on the occasion of the funeral of Prince Willem the Silent. Saenradam left behind a large quantity of sketches and preparatory drawings.

Houckgeest had been trained as an imaginary architectural painter in all probability by Bartolomeus van Bassen (1590 - 1652) who started his career in Delft and lived nearby in Hague.1 Throughout his career Van Bassen stayed faithful to the subject of imaginary architecture.2 Constructing artificial space excludes the use of the camera obscura and he could possibly achieve such a precise space only with geometry. The infrared reflectogram shows that Van Bassen made a precise constructional underdrawings.

Van Bassen's contemporary, Pieter Anthonisz van Bronckhorst (1588 - 1661), was also born in Delft and excelled in depicting the church interiors and temples, which he supplemented with small figures.3

Another Delft born Dirck van Delen (c. 1605 - 1671) was a painter who specialized exclusively in architectural paintings, principally palace perspectives and church interiors. Delen's pupils were Daniël de Blieck (c. 1610 - 1673) and Hans Jurriaensz van Baden who both specialized in architectural paintings and real or imaginary church interiors.4

All these masters were preceded by Hans Vredeman de Vries (1527 - c. 1607), a Dutch Renaissance architect, painter, and engineer. Although his viewing angles are in many cases too wide, he worked with the principles of The Renaissance perspective system. His son Paul trained and cooperated with Hendrick Aerts.

One of the artists who came to Delft and strengthened the new focus on realistic representation of the inner space was Pieter de Hooch. He arrived at about 1655 and was probably more influential to Vermeer than anybody else. Despite the close association of style between Vermeer and De Hooch their emotionality was different. Vermeer focused on the psychological moods, perhaps due to his introvert nature, his figures seem to be caught in the moment of contemplation. Vermeer's brushwork was more precise and his composition very carefully thought out.

To the list of above mentioned painters, who knew and commanded the Renaissance perspective system, belongs also Cornelis de Man, Jan Steen, and Amsterdam's Samuel Hoogstraten and Pieter Janssens Elinga.

Yet even though they all used the same system, their practical working procedure could differ. Bartolomeus van Bassen made some constructional underdrawings directly on the canvas. Vermeer, on the other hand, had the design ready in advance. Seanredam made preparatory sketches but did not include the main constructional lines in it. Once the painting was finished the preparatory constructions was destroyed.

JØrgen Wadum had proposed that Vermeer used a chalked string stretched from a pin inserted at the vanishing point. He would have then plucked the string to mark chalk lines. In terms of practical use, the method of chalked strings is inaccurate and hardly applicable. Eventually why would any painter soak a string in a chalk and then nail and stretch it to a canvas when he could make a line more accurately and much quicker with a ruler, either by the crayon or even chalk itself, right on the canvas? One will soon realize that this method is impractical and does not address the location of lateral points.

The constructional scheme goes well beyond the borders of the canvas and small scale of Vermeer's pictures requires the high precision which is impossible to attain by strumming some strings. Vermeer made all the preliminary preparations on a sheet of thin paper, which he latter pinned to the canvas at the vanishing point and then transferred by the pouncing method to the canvas.  

Three Point Perspective

The principle of a perspective scheme is generally misunderstood and it is apparent that the church painters in Delft or any perspective masters did not distinguish the two-point perspective from one-point perspective as it is presented to us today. The perspective system always had three-points, either linear or the Renaissance Perspective. The basis of the standard three-point construction is the central vanishing point and two lateral points lying on the horizon. The lateral points are mistakenly called the distance points. I have to emphasize that the lateral point is not the same as the distance point and is not equal to the original viewing distance.

The whole principle of perspective representation depends on how to determine the scope of foreshortening. For simpler visualization let us imagine a cube or square. If we view a cube or square en face and then turn it to the edge, will the foreshortening method change? Of course not. It is just the object viewed from the edge; an object laid differently against our view. In order to foreshorten the subject, we need the vanishing point and also the lateral points. So, when we look at the cube's front side we still need the lateral points to determine where to put the back wall of the cube. And this applies to any other objects. The linear perspective uses an estimate, whereas the Renaissance perspective system uses a method that determines how to foreshorten the orthogonals converging in the vanishing point.

FIG.1/ Three point perspective

Do you see 6 trapezoids or 8 rhombuses in the picture? Is it a two-point or one-point perspective?

It is a three-point perspective and there are 14 squares. 6 of them we view from the front side and 8 of them from the edge. Those from the edge are √2 smaller. To create one or the other we need all three points and it is purposeless to divide the perspective into one or two pointed.

Jean Pelerin in 1505 published a book called De Artificiali perspectiva, which is filled with many examples of perspective rendering in practice. Skillfully drawn examples are accompanied by a short and concise text that is free of philosophical gibberish. The basis of its construction is a three-point system with a central vanishing point and two lateral points at either side, which he calls 'principal point' and 'tiers points'. "These tiers points he rather confusingly attributes to the turning of the 'diametr' of the visual pyramid. Pelerin suggests that the production of the apex of the visual pyramid in the eye may be compared to the focusing effect of a 'burning (concave) mirror'. He also states that a 'third point' needs to be moved further apart as the spectator's distance increases".5 (Kemp 66)

In a few sentences Pelerin outlines the foundation of the Renaissance perspective system.

Yet he leaves the subject incomplete and does not explore or explain any further.

The renaissance masters held the golden cut in high esteem and many painters draw attention to it in their works. There are few ways how to construct tiers point geometrically, but we can interpret Pelerin's description as follows.  

FIG.2/ Focal length point - apex of visual pyramid

The apex of the visual pyramid in the eye with the focusing effect of a 'burning mirror' is positioned at the point F. The F point lies at the bottom of a √1.618 circle and in spite of its great importance it is not the viewer's point. If we use modern terminology, the F point is rather a focal length point.6 It is an essential element to determine the tiers points, which we obtain by the turning of the 'diameter'. By turning compasses we can locate the tiers points T1, T2 on the sides. And the distance points to be moved further apart. How far and the relationship between the lateral point and the distance point will be explained in the chapter The Art of Painting.

The Secret of The Old Masters

Do not assume that the knowledge of the Renaissance Perspective was accessible or automatically transmitted from teacher to pupil. It was a valuable know-how. Nevertheless, with such a rapid development of perspectival representation, I would suppose that some information transmission must have occurred.

It certainly was not Emanuel de Witte who revealed the secret system to his pupils, he was so protective that he even shouted at people watching him at work in church. There were pinholes found in his work too and there is no evidence that he used the camera obscura.

On the other hand, both Delen's pupils, Daniël de Blieck and Hans Jurriaensz van Baden, knew the method and it is obvious that the master had passed on the knowledge to his apprentices.

Likewise Hans Vredeman de Vries probably passed on some knowledge to his pupil Van Steenwijck and he later to his son Hendrik van Steenwijk.7

The only Dutch painter who is documented to have used building's measurements to construct a perspective scheme is Pieter Saenredam. Saenredam on the basis of precise measurement made perspectival drawings which he transferred on his canvas. To preserve the fidelity of the image he had to made accurate measurements of the church interiors, which he then applied to his ground plans. He began his composition drawings by establishing the so-called 'eye', which indicated the viewing point on the horizon.8 Only the meticulous geometrical preparation helped him to create such accurate and captivating paintings.

Architect Jacob van Campen (1596- 1657) friend of Saenredam, lived in Italy 1616 - 1624,9 also knew the system. He could have influenced Saenredam after returning to the Netherlands, he might have passed on some or partial knowledge to his friend. It could not be his teacher Frans Pieter de Grebber, because he was not knowledgeable of the perspective system. Saenredam's large personal library, contained only two books on perspective, one by Serlio the other by Dürer. Martin Kemp asserts that Serlio's text did not discuss the distance point method while Dürer described it in an "almost incomprehensible and partially incorrect" manner.10 Kemp is right with Dürer, who endeavored to figure out the system his whole life. But he is mistaken with Serlio and marks his method of the viewing distance as wrong. Serlio treated the distance point method intentionally very vaguely. He does not explain and develop any instructions; one has to work it out and link it to the whole concept. On one page he illustrates how to create a tiers points in one chart and how to create a distance point in the other. If we combine these two graphs, we get the base of the Renaissance system.

FIG. 3/ Distance points - the Renaissance perspective system

Vermeer's father was an art dealer; he registered as such in the Delft Guild of Saint Luke a few years after Bartolomeus van Bassen checked out and moved to The Hague.11 Vermeer's father certainly knew van Bassen's work and must have been talking to his son about perspective.

Bartolomeus van Bassen was first Dutch architect/painter to specialize in the genre of architectural painting and played pivotal role in the development of painting in this field. He achieved precise perspectival space in his imaginary paintings only through the use of geometry. All this suggests that Bartolomeus van Bassen and perhaps Pieter Anthonisz van Bronckhorst planted a seed that has grown into the Delft's unique branch of the Dutch Golden Age.

On the infrared image of Interior of Church with a Procession, we can see some preparatory constructional underdrawings left behind by Bartholomeus van Bassen. The detail examination of this infrared reflectogram clearly shows the outline of the pillar's foot and orthogonal lines of the stairs behind.

FIG. 4/ Infrared reflectogram detail of Bartholomeus van Bassen, Interior of a Church with a Procession, Statliche Museen zu Berlin.

Little above, the horizontal line and the diagonal line meet at the edge of the pillar. These lines are remains of the constructional geometry and to decipher what they represent we need to reconstruct at least the basic scheme of the perspective system Bartholomeus van Bassen worked with.

Like any other painter using the Renaissance Perspective system, Bartholomeus van Bassen too divided the space into the basic squares (blue squares).Fig.5 This square division was sometimes called quadratura ; a term introduced in the seventeenth century became popular with Baroque artists. The term is most-commonly associated with Italian ceiling painting, but it can also mean the "opening up" of walls through architectural illusion. It is referring to a spatial division of foreshortened squares and is directly tied to seventeenth-century theories of perspective and the representation of architectural space.12

As in most of his paintings van Bassen used the golden ratio as a default paradigm. All three circles adapt the proportions of the golden ratio. The first circle serves as an intermediate stage to form the other two circles necessary for the spatial construction.

The horizontal line we see in the infrared image is the upper line of the fourth square (green line). The line also determines the bottom edge of the side chapel walls.The green diagonal line begins at point A passes through point B at the picture plane and meets the circle at the lateral point T1 on the horizon. The line from vanishing point V to point A, which runs along the first step, determines the width of the nave. The width of the nave to the base square (blue) accordingly reflects the golden mean proportion. Fig.5

FIG. 5/ Bartholomeus van Bassen, Interior of a Church with a Procession, 1624, with figures by Frans Francken II, signed by both artists. © Statliche Museen zu Berlin. Biltarchiv Preussicher Kulturbesitz. Photo: Jorg P. Anders. Inv.695 

To make the deciphering easier van Bassen indicated (or instructed his staffage associate to indicate) the elementary points of the structure by putting the long burning candles and beggar's stick to the hands of the figures in the front. The candles and stick (red lines) aim exactly to the points which correspond to the golden ratio components. I do not think his intention was to provide us with a hint, but to facilitate his work and orient himself in the construction process by pointing out the important components of the spatial design. The red lines (which follow the candle sticks) start at the points defined by the golden ratio circles and aim to the corners or the centers of the foreshortened squares.

Moreover, rectangular panels on the edge of the picture plane too suggest the picture's golden ratios proportions. One orthogonal leading along the panel's side points to the intersection of the golden ratio circle and the base line; the other points to the corner of the base square. To emphasize the geometric importance of the base square, van Bassen puts the first columns on its edges.

Note also that the distance of the upper ledge above the nave arcs to the horizon corresponds to the ratio √2 to 1

Vermeer's Perspective

Vermeer registered as a master painter in the Guild of Saint Luke on 29 December 1653. It is not known who taught him to paint. It has been proposed that either Carel Fabritius or Gerard de Borch was his teacher.15 But not all painters in Delft possessed the knowledge of the Renaissance Perspective and these two great painters were among them. The apparent lack of acquaintance with the rules of perspective in Vermeer's early works suggests that he did not learn this skill during his apprenticeship but figured it out latter on his own. He was inspired by a group of painters working in Delft and near The Hague who have mastered the use of the Renaissance perspective and have successfully applied it in practice. It would be unnatural, if in a city as small as Delft, highly talented painters belonging to the same guild would not talk about their craft and closely examine the progress of their colleagues.

It appears that Vermeer figured out the Renaissance Perspective system for himself in a period of a low painterly activity between 1657 - 1660. During this period he must have spent countless hours handling rulers and compasses. We can see him with compasses in his hand looking out of window in his self-portrait The Geographer as he is working on the perspective design for The Little Street. The window he is looking out is 44cm wide; and 4 out of 6 window's dividers measure 54cm. Vermeer is thus telling us that he is looking at the scenery of The Little Street, whose dimensions are 54 x 44cm.

Reconstructing the The Little Street we can find out that the viewing point is distanced around 12 meters from the building we are looking at. The distance between buildings across the canal at the Voldersgracht street measures 11 meter,16 and if we add an another meter we arrive at Vermeer's eye in The Geographer.

The Geographer or The Astronomer could be Vermeer's lost self-portrait, item 3 from Dissius auction, which is now presumed missing.

Benjamin Binstock suggests that Vermeer used his family members as models and depicted his pregnant wife in several cases.17 He too endorses The Geographer and The Astronomer asVermeer'sself-portraits.

There is no written historical document reference which would provide instance that Vermeer did use his family members as models. Yet, at least in Woman with a Balance and Woman in Blue Reading a Letter, we can observe a woman in an advanced stage of pregnancy. Vermeer had 11 children and his wife, Catharina Bolnes, must have been pregnant for much of her married life time. Sincepregnancy was not a commonly portrayed scene inDutchpainting of the seventeenth century, it is self-evident that this was not commissioned work and the subject of his art was his expectant wife. It was Vermeer's externalization of love and affection expressed internally within the walls of his home.

Binstock also says that some of Vermeer work "display qualities unknown in Vermeer: imbalance and imperfections, technical difficulties and glaring mistakes, distortions, a lack of an interior, relative darkness or blurred definition"(Binstock, 2017). He attributed these deficiencies to the possibility that Vermeer's eldest daughter had painted several paintings herself. I think these apparent shortcomings are due to the fact that Vermeer's work is viewed from the wrong chronological angle.

Dating

First and foremost, Vermeer works should be ordered chronologically as accurately as possible, at least within our capabilities. Certain development in Vermeer's work can be detected, not only in technique, but also in an ability to capture the moment. Yet, it is a mystery to me how leading experts could agree so harmoniously on all dates in such a diverse work area with all these inconsistencies?

How could be The Girl with a Red Hat and Young Girl with a Flute dated 1667, the same year as The Art of Painting, when it bears so many technical mistakes and imperfections. Moreover, it is painted on the wood, a type of material on which young artists try to paint their first works. One must wonder how the evaluation process was conducted.

Only three of Vermeer's paintings were dated by the artist. In the light of this, I should like to suggest that dates of paintings such as Girl Reading a Letter at an Open Window, Little street or Officer and Laughing Girl should be estimated after 1660. All these paintings bear advanced knowledge of the Renaissance perspective and are estimated consecutively to 1657, 1657-58, 1658.

The creation date of Girl Interrupted at her Music is proposed by the leading experts at around 1660. Yet its composition does not contain the trace of the Renaissance perspective system. In this regard, the paintings created before this date should not show signs of the Renaissance Perspective system. These three paintings mentioned above should be dated above Girl Interrupted at her Music.

The Music Lesson, because it is created with an elementary knowledge of the Renaissance perspective system, should be at the beginning of the line of his perspectival works. The scheme used at The Music Lesson is more suitable for larger vast spaces and it is likely that Vermeer studied and eventually deciphered the system by analyzing the works of church interior painters.

The Music Lesson

Vermeer was praised by his contemporary art lovers for his mastery of perspective, he was a master of capturing moments, achieving better effects than any camera can do today. Many people tried to copy his painting technique, but the results were always unconvincing. It is the sacred geometry and the Renaissance perspective system that stands behind the eternal effect of his paintings.
It was his painting The Music Lesson where I first realized the presence of the Golden Ratio.

Yet, this divine proportion is employed not on a surface, but in-depth or three-dimensionally.   

FIG. 7/ The Music Lesson © Royal Collection, London

Vermeer is so kind to inform us, although implicitly, about the presence of the golden ratio (green triangle). The floor and the wall create the hypotenuse of the golden ratio triangle which inclines under the 51.8/38.2 degrees. Using the circles Vermeer determined the focal lenght distance. From the F point, given by the golden ratio, he then constructed the lateral points. However, the lateral points do not reach the distance given by the golden ratio. This is because Vermeer, similarly like van Bassen, used orthogonals which do not lead from the corners, but from the intersection of the circle and the side of the bottom square.

As I have mentioned earlier the tiers or lateral points play an essential role in the system and define the foreshortened square or a back wall of the cube. The base square and subsequently the cube provide us with the exact measurement to work with the three-dimensional space of the picture.

The first step in the system's sequence is to transfer the rectangular area of the picture into the square. From this base square we will advance to the construction of two circles. One circle with the ratio of radius to half square side equals √ φ to 1, and one with the ratio of radius to half square side equals φ to 1. It is worth mentioning that the larger φ circle passes through the golden ring on which the mirror is hung. Fig.7

Notice that the F point is little off-axis. This is because Vermeer has slightly moved, deliberately or not, the sheet of paper with preparatory scheme around the pin in the opposite direction. This causes the ceiling beams tilt down to the right and the bottom right tile to be cut off.

Constructional scheme of The Music Lesson follows the principle of Pelerin's apex of the visual pyramid at the point F. Fig.2

The room consist of 16 tiles, which equals 6.4 meters in terms of length and 3 meters in terms of height (up to the wooden beams). Table is around 90 cm high. Virginal measures around 168 cm, which corresponds to the standard size of Ruckers virginal from the 17 century.18 All these calculations are done with The Rule of Three19 and we can thus find out size of any object in the picture. Calculations are carried out provided that the tile measures 28.3 cm.

I derived the size from smaller 17th century Dutch Delft tiles, which are sold until today.20 They measure 13 by 13 centimeters and occurs in few of the Vermeer's paintings.

FIG.8/ Detail of Lady standing at a Virginal, National Gallery, London

These tiles defended plaster against regular attacks of the mops and prevented wall humidification. We can see them in the cutout of Lady standing at a Virginal below. If we know that the size of a smaller tile is 13cm then with the help of The Rule of Three we arrive at around 40 cm for diagonal of a larger tile. Resulting side of a single tile then comes to 28.3 cm, which is the value corresponding to Amsterdam's voet at that time. The exact size of a voet (foot) depended on the length of the local voet, which changed from region to region. The most commonly used voet in the Netherlands was the Rijnland foot, which was between 28 -31cm.21  

In the mirror reflection we can see his easel and a sheet of paper rolled on the floor. Next to the easel on the last white tile we can see also a dark stick. This could be a chair's leg or a mahlstick as a painter puts it on the floor and watches the scene in front of him. 

 Upon further careful observation of the mirror image, we find that Vermeer missed a chair. The confrontation of the mirror image with the side and floor plan only confirms the absence of the upper part of the chair. Fig.10 This fact merely manifests that he did not use a camera obscura and did not slavishly copied its projected image. He may have forgotten, but more likely it did not suit his composition.

FIG. 9/ Detail of The Music Lesson, Royal Collection, London

Side view analysis reveals that mirror reflection (green lines) starts at the edge of the picture plane and hence we can confirm our count of 16 tiles from wall to wall. If we follow the law of reflection, which states that the angle of incidence is equal to the angle of reflection, then we will find the correct tilt of the mirror 

FIG. 10/ The Music Lesson - side view with the viewpoint at 14.5th tile

Professor Philip Steadman in his book Vermeer's Camera claims that Vermeer had used a camera obscura to create his paintings. Steadman found six of his paintings that are precisely the right size as if they had been painted from inside a camera obscura in the room's back wall. I closely examined his constructional drawing to The Music Lesson and my conclusion may surprise some.

Steadman's number of tiles in the The Music Lesson likewise corresponds to the count of 16 tiles throughout the room and his estimate of the tile is 29,3 cm. The position at which Vermeer would have put his eye Steadman labeled also at 14.5th tile.22

In the case of the viewing point at the 14.5th tile the size of the projected image on the back wall would be only about 58cm vertically and 50cm horizontally. The dimensions of the painting is 74.6 x 64.1 cm. It still differs from Steadman's claim of a full 16cm lengthwise and 14cm crosswise. Fig. 10 and Fig. 11

FIG. 11/ The Music Lesson - floor plan

Upon careful examination of Fig.9 and Fig. 10, we find out that there are are some inconsistencies in the mirror reflection. Vermeer appears to have reflected the light beams (green lines) from the right side of the mirror. As he supposed to do it from both sides.

Vermeer pivotal perspectival paintings such as The Art of Painting, The Allegory of Faith or The Love Letter are not included in Steadman's selection of six paintings.

The viewpoint of The Love Letter sits on 19th tile from the back wall. Moreover, here Vermeer would have to move the camera obscura apparatus to a narrow, dark hall.

The viewpoint of The Allegory of Faith sits also behind 15th tile from the back wall, yet its viewing angle is much narrower than the The Music Lesson and its image projected on the wall would be even smaller than 37 x 32cm, while its real dimensions are 114 x 99cm.

The viewpoint of The Art of Painting is on 14th tile.

Analyses of The Art of Painting, The Allegory of Faith or The Love Letter will follow in part 2.

continues in part 2.

 REFERENCES

1. Liedtke, W. and Liedtke, W. (2001). Vermeer and the Delft school. New York: The Metropolitan Museum of Art.

2. Rüger, A. and R, Billinge. (2005) The Design Practices of the Dutch Architectural Painter Bartholomeus van Bassen, The National Gallery Technical Bulletin. National Gallery.

3. "Pieter Anthonisz van Bronckhorst." Wikipedia, Wikimedia Foundation, 2019, en.wikipedia.org/wiki/Netherlands_Institute_for_Art_History.

4. "Dirck van Delen." Wikipedia, Wikimedia Foundation, 2019, en.wikipedia.org/wiki/Netherlands_Institute_for_Art_History.

5. Kemp, M. The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. Yale University Press, 1990.p.66

6. Bouc, P. (20019).www.renaissanceperspective.cz

7. Yann, P. and Dutoit, D. "Hendrik Van Steenwijk I." Hendrik Van Steenwijk i : Définition De Hendrik Van Steenwijk i Et Synonymes De Hendrik Van Steenwijk i (Anglais), dictionnaire.sensagent.leparisien.fr/hendrik van steenwijk i/en-en/.

8. Helmus, Liesbeth M., and Arie de. Groot. Pieter Saenredam: the Utrecht Work: Paintings and Drawings by the 17th-Century Master of Perspective. J. Paul Getty Museum, 2002.

9. Swillens, P. T. A., and P. T. A. Swillens. Jacob Van Campen: Schilder En Boumeester, 1595-1657. Gysbers & Van Loon, 1979.

10. Kemp, M. The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. Yale University Press, 1990.

11. Rüger, A. and R, Billinge. (2005) The Design Practices of the Dutch Architectural Painter Bartholomeus van Bassen, The National Gallery Technical Bulletin. National Gallery.

12. Wittkower, R. and Connors, J and Montagu, J. Art and Architecture in Italy, 1600-1750, vol. 1, Pelican history of art, New Haven: Yale University Press (1999): 35-36.

13. "Spinoza and Diamond Polishing?" Frames /Sing, 30 July 2008, kvond.wordpress.com/2008/07/30/spinoza-and-diamond-polishing/.

14. Tolkowsky, M. Diamond Design: a Study of the Reflection and Refraction and Refraction of Light in a Diamond (Classic Reprint). FORGOTTEN Books, 2015.

15. Wheelock, A. Johannes Vermeer. Harry N. Abrams, 1998.

16. https://www.essentialvermeer.com/delft/little-street-steadman/little-street-steadman.html#.XTngZOgzZhE

17. Binstock, B. (2017) Interiors and interiority in Vermeer: empiricism, subjectivity, modernism. Palgrave Communications. 3.17068 doi: 10.1057/palcomms.2017.68.

18. Metmuseum.org, www.metmuseum.org/art/collection/search/505004.

19. Baxandall, M. Painting&Experience in Fifteenth-Century Italy, Oxford University Press, 1988 p. 95,

20. Regts - Antique Tiles, www.antiquetileshop.com/faq.

21. "De Oude Nederlandse Maten En Gewichten." Maten En Gewichten, www.meertens.knaw.nl/mgw/.

22. Vermeers Camera, Uncovering the Truth Behind the Masterpieces. A Book by Philip Steadman, www.vermeerscamera.co.uk/room5.htm.