This article looks at some of the interactions between mathematics and art in western culture. There are other topics which will look at the interaction between mathematics and art in other cultures. Before beginning the discussion of perspective in western art, we should mention the contribution by al-Haytham. It was al-Haytham around 1000 A.D. who gave the first correct explanation of vision, showing that light is reflected from an object into the eye. He studied the complete science of vision, called perspectiva in medieval times, and although he did not apply his ideas to painting, the Renaissance artists later made important use of al-Haytham's optics.

There is little doubt that a study of the development of ideas relating to perspective would be expected to begin with classical times, and in particular with the ancient Greeks who used some notion of perspective in their architecture and design of stage sets. However, although Hellenistic painters could create an illusion of depth in their works, there is no evidence that they understood the precise mathematical laws which govern correct representation. We chose to begin this article, therefore, with the developments in the understanding of perspective which took place during the Renaissance. First let us state the problem: how does one represent the three-dimensional world on a two-dimensional canvass? There are two aspects to the problem, namely how does one use mathematics to make realistic paintings and secondly what is the impact of the ideas for the study of geometry.

By the 13^th Century Giotto was painting scenes in which he was able to create the impression of depth by using certain rules which he followed. He inclined lines above eye-level downwards as they moved away from the observer, lines below eye-level were inclined upwards as they moved away from the observer, and similarly lines to the left or right would be inclined towards the centre. Although not a precise mathematical formulation, Giotto clearly worked hard on how to represent depth in space and examining his pictures chronologically shows how his ideas developed. Some of his last works suggest that he may have come close to the correct understanding of linear perspective near the end of his life.

The person who is credited with the first correct formulation of linear perspective is Brunelleschi. He appears to have made the discovery in about 1413. He understood that there should be a single vanishing point to which all parallel lines in a plane, other than the plane of the canvas, converge. Also important was his understanding of scale, and he correctly computed the relation between the actual length of an object and its length in the picture depending on its distance behind the plane of the canvas. Using these mathematical principles, he drew two demonstration pictures of Florence on wooden panels with correct perspective. One was of the octagonal baptistery of St John, the other of the Palazzo de Signori. To give a more vivid demonstration of the accuracy of his painting, he bored a small hole in the panel with the baptistery painting at the vanishing point. A spectator was asked to look through the hole from behind the panel at a mirror which reflected the panel. In this way Brunelleschi controlled precisely the position of the spectator so that the geometry was guaranteed to be correct. These perspective paintings by Brunelleschi have since been lost but a "Trinity" fresco by Masaccio from this same period still exists which usesBrunelleschi's mathematical principles. 
Here is a picture of Masaccio's Holy Trinity
It is reasonable to think about how Brunelleschi came to understand the geometry which underlies perspective. Certainly he was trained in the principles of geometry and surveying methods and, since he had a fascination with instruments, it is reasonable to suppose that he may have used instruments to help him survey buildings. He had made drawing of the ancient buildings of Rome before he came to understand perspective and this must have played an important role.

Now although it is clear that Brunelleschi understood the mathematical rules involving the vanishing point that we have described above, he did not write down an explanation of how the rules of perspective work. The first person to do that was Alberti in his treatise On painting. Now in fact Alberti wrote two treatises, the first was written in Latin in 1435 and entitled De pictura while the second, dedicated to Brunelleschi, was an Italian work written in the following year entitled Della pittura. Certainly these books are not simply the same work translated into two different languages. Rather Alberti addresses the books to different audiences, the Latin book is much more technical and addressed to scholars while his Italian version is aimed at a general audience.

De pictura is in three parts, the first of which gives the mathematical description of perspective which Alberti considers necessary to a proper understanding of painting. It is, Alberti writes:-

... completely mathematical, concerning the roots in nature from which arise this graceful and noble art.

In fact he gives a definition of a painting which shows just how fundamental he considers the notion of perspective to be:-

A painting is the intersection of a visual pyramid at a given distance, with a fixed centre and a defined position of light, represented by art with lines and colours on a given surface.

Alberti gives background on the principles of geometry, and on the science of optics. He then sets up a system of triangles between the eye and the object viewed which define the visual pyramid referred to above. He gives a precise concept of proportionality which determines the apparent size of an object in the picture relative to its actual size and distance from the observer.

One of the most famous examples used by Alberti in his text was that of a floor covered with square tiles. For simplicity we take the centric point, as Alberti calls it (today it is called the vanishing point), in the centre of the square picture. 

Here is a Alberti's construction of perspective for a tiled floor

In our diagram the centric point is C. The square tiles are assumed to have one edge parallel to the bottom of the picture. The other edges which in reality are perpendicular to these edges, will appear in the picture to converge to the centric point C. The diagonals of the squares will all converge to a point D on a line through the centric point parallel to the bottom of the picture. The length of CD determines the correct viewing distance, that is the distance the observer has to be from the picture to obtain the correct perspective effect. Alberti chooses not to give mathematical proofs, however, writing:-

We have talked as much as seems necessary about the pyramid, the triangle, the intersection. I usually explain these things to my friends with certain tedious geometrical proofs, which in this commentary it seems to me better to omit for the sake of brevity.

Pictures from this period which include a square tiled floor are called pavimento (Italian for floor) pictures. There are many examples of such pictures in the years following Alberti's book which had a huge influence on painting.

Of course the pavimento provides a type of Cartesian coordinate system. Alberti shows how to use the grid to obtain the correct shape for a circle. Place a circle on a square grid and mark where the squares cut the circle. Construct the perspective view of the square grid as above and reconstruct the circle by seeing the positions of the points of intersection in the projected view. The circle will project into an ellipse, but it would be a long time before the importance of projecting conic sections was realised.

Next we should mention Lorenzo Ghiberti who was born in Pelago, Italy around 1378. He is famed as a sculptor and his most famous work is the bronze doors on the east side of the baptistery in Florence. He created two sets of doors and before he designed the second set he had become familiar with the new ideas on perspective as set out by Alberti. The doors contain ten panels which, Ghiberti wrote, exhibit:-

... architectural settings in the relation with which the eye measures them, and real to such a degree that ... one sees the figures which are near appear larger, and those that are far off smaller, as reality shows it.

Ghiberti is also important for his treatise I Commentarii, written around 1447, in three volumes. The work contains a history of art in ancient times, a history of thirteenth century artists, an autobiography, and a compilation of medieval texts on the theory of vision such as that by al-Haytham. This was important since, as we mentioned at the beginning of this article, al-Haytham and others had studied optics and vision without relating the ideas to painting, while now Ghiberti showed the relevance of the earlier ideas on optics to art.

The most mathematical of all the works on perspective written by the Italian Renaissance artists in the middle of the 15^th century was by Piero della Francesca. In some sense this is not surprising since as well as being one of the leading artists of the period, he was also the leading mathematician writing some fine mathematical texts. In Trattato d'abaco which he probably wrote around 1450, Piero includes material on arithmetic and algebra and a long section on geometry which was very unusual for such texts at the time. It also contains original mathematical results which again is very unusual in a book written in the style of a teaching text (although in the introduction Piero does say that he wrote the book at the request of his patron and friends and not as a school book). Is there a connection with perspective? Yes there is, for Piero illustrates the text with diagrams of solid figures drawn in perspective.

Here is Piero's illustration of a dodecahedron 

Continuing the theme of the regular solids, we note that a later text by Piero is Short book on the five regular solids. However, it is his three volume treatise On perspective for painting (some believe written in the mid 1470s, others believe written in the 1460s) which is of most interest to us in this article. His book begins with a description of painting:-

Painting has three principal parts, which we say are drawing, proportion and colouring. Drawing we understand as meaning outlines and contours contained in thing. Proportion we say is these outlines and contours positioned in proportion in their places. Colouring we mean as giving the colours as shown in the things, light and dark according as the light makes them vary. Of the three parts I intend to deal only with proportion, which we call perspective, mixing in with it some part of drawing, because without this perspective cannot be shown in action; colouring we shall leave out, and we shall deal with that part which can be shown by means of lines, angles and proportion, speaking of points, lines, surfaces and bodies.

We see from this introduction that Piero intends to concentrate on the mathematical principles. Perhaps it is most accurate to say that he is studying the geometry of vision which he later makes clearer:-

First is sight, that is to say the eye; second is the form of the thing seen; third is the distance from the eye to the thing seen; fourth are the lines which leave the boundaries of the object and come to the eye; fifth is the intersection, which comes between the eye and the thing seen, and on which it is intended to record the object.

Piero begins by establishing geometric theorems in the style of Euclid but, unlike Euclid, he also gives numerical examples to illustrate them. He then goes on to give theorems which relate to the perspective of plane figures. In the second of the three volumes Piero examines how to draw prisms in perspective. Although less interesting mathematically than the first volume, the examples he chooses to examine in the volume are clearly important to him since they appear frequently in his own paintings. The third volume deals with more complicated objects such as the human head, the decoration on the top of columns, and other "more difficult shapes". For this Piero uses a method which involves a very large amount of tedious calculation. He uses two rulers, one to determine width, the other to determine height. In fact he is using a coordinate system and computing the correct perspective position of many points of the "difficult shape" from which the correct perspective of the whole can be filled in.

Piero della Francesca's works were heavily relied on by Luca Pacioli for his own publications. In fact the third book of Pacioli's Divina proportione is an Italian translation of Piero's Short book on the five regular solids. The illustrations in Pacioli's work were by Leonardo da Vinci and include some fine perspective drawings of regular solids.

Here is a Leonardo's illustration

Now in Leonardo's early writings we find him echoing the precise theory of perspective as set out by Alberti and Piero. He writes:-

... Perspective is a rational demonstration by which experience confirms that the images of all things are transmitted to the eye by pyramidal lines. Those bodies of equal size will make greater or lesser angles in their pyramids according 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.

He developed mathematical formulas to compute the relationship between the distance from the eye to the object and its size on the intersecting plane, that is the canvas on which the picture will be painted:-

If you place the intersection one metre from the eye, the first object, being four metres from the eye, will diminish by three-quarters of its height on the intersection; and if it is eight metres from the eye it will diminish by seven-eighths and if it is sixteen metres away it will diminish by fifteen-sixteenths, and so on. As the distance doubles so the diminution will double.

Not only did Leonardo study the geometry of perspective but he also studied the optical principles of the eye in his attempts to create reality as seen by the eye. By around 1490 Leonardo had moved forward in his thinking about perspective. He was one of the first people to study the converse problem of perspective: given a picture drawn in correct linear perspective compute where the eye must be placed to see this correct perspective. Now he was led to realise that a picture painted in correct linear perspective only looked right if viewed from one exact location. Brunelleschi had been well aware of this when he arranged his demonstration of perspective through a hole. However for a painting on a wall, say, many people would not view it from the correct position, indeed for many paintings it would be impossible for someone viewing them to have their eye in this correct point, as it may have been well above their heads.

Leonardo distinguished two different types of perspective: artificial perspective which was the way that the painter projects onto a plane which itself may be seen foreshortened by an observer viewing at an angle; and natural perspective which reproduces faithfully the relative size of objects depending on their distance. In natural perspective, Leonardo correctly claims, objects will be the same size if they lie on a circle centred on the observer. Then Leonardo looked at compound perspective where the natural perspective is combined with a perspective produced by viewing at an angle. Perhaps in Leonardo, more than any other person we mention in this article, mathematics and art were fused in a single concept.

The story we have told up to this point has been very much an understanding of perspective in Italy by artists and mathematicians learning personally from each other. By 1500, however, Dürer took the development of the topic into Germany. He did so only after learning much from trips to Italy where he learned at first hand from mathematicians such as Pacioli. He published Unterweisung der Messung mit dem Zirkel und Richtscheit in 1525, the fourth book of which contains his theory of shadows and perspective. Geometrically his theory is similar to that of Piero but he made an important addition stressing the importance of light and shade in depicting correct perspective. An excellent example of this is in the geometrical shape he sketched in 1524.

Here is a Dürer's shaded geometrical design

Another contribution to perspective made by Dürer in his 1525 treatise was the description of a variety of mechanical aids which could be used to draw images in correct perspective.

Let us consider a number of other contributions to the study of perspective over the following 200 years. We mention first Federico Commandino who published Commentarius in planisphaerium Ptolemaei in 1558. In this work he gave an account of Ptolemy's stereographic projection of the celestial sphere, but its importance for perspective is that he broadened the study of that topic which had up until then been concerned almost exclusively with painting. Commandino was more interested in the use of perspective in the making of stage scenery principally because his main interest was in classic texts and, unlike many earlier treatises he was writing for mathematicians rather than artists.

Wentzel Jamnitzer wrote a beautiful book on the Platonic solids in 1568 called Perspectiva corporum regularium. This is not a book designed to teach perspective drawing but, nevertheless, contains many illustrations superbly drawn in perspective. He is clear in his intention:-

All superfluity will be avoided and, in contrast to the old fashioned way of teaching, no line or point will be drawn needlessly.

Daniele Barbaro's La Practica della perspectiva published in 1569, the year after Jamnitzer's treatise, complained that painters had stopped using perspective. Taken at face value this is not true, but what he undoubtedly meant was that painters were not painting architectural scenes. Barbaro was interested in perspective in stage sets mainly because he had published an Italian translation of Vitruvius's On architecture in 1556 and his interest had been aroused by this work. His 1569 treatise shows that he had studied the work of Piero and Dürer carefully and the methods he gave for perspective constructions were variations on their methods.

Egnatio Danti, like so many of the others we have mentioned in this article, was both an excellent mathematician and artist. His preface to Le due regole della prospettiva pratica di M Iacomo Barozzi da Vignola was published in 1583, three years before his death. In his introduction to this work Danti wrote a brief history of perspective:-

... we know of no book or written document which has come down to us from ancient practitioners, although they were mot excellent, as is convincingly shown by the descriptions of the stage scenery they made, which was much prized both in Athens among the Greeks and in Rome among the Latins. But in our own time, among those who have left something of note in this art, the earliest, and one who wrote with best method and form, was Messer Pietro della Francesca dal Borgo Sansepolcro, from whom we have today three books in manuscript, most excellently illustrated; and whoever wants to know how excellent they are should look to Daniele Barbaro, who has transcribed a great part of them in his book on Perspective.

Not only did Danti write an introduction to his edition of Vignola's treatise, but he also added considerably to its content by giving mathematical justification where Vignola simply states a rule to be applied.

The next contributor we mention is Giovanni Battista Benedetti who was a pupil of Tartaglia. He produced a work entitled A book containing various studies of mathematics and physics in 1585 which contains a treatise on arithmetic, some other short works and letters on various scientific topics, as well as a short treatise on perspective De rationibus operationum perspectivae. In his perspective treatise Benedetti was concerned not just with rules for artists working in two dimensions but with the underlying three-dimensional reasons for the rules.

We mentioned Commandino above and the next person who we want to note for his contribution to perspective, Guidobaldo del Monte, was a pupil of Commandino. Del Monte's six books on perspective Perspectivae libri sex (1600) contain theorems which he deduces with frequent references to Euclid'sElements. The most important result in del Monte's treatise is that any set of parallel lines, not parallel to the plane of the picture, will converge to a vanishing point. This treatise represents a major step forward in understanding the geometry of perspective and it was a major contribution towards the development of projective geometry.

In 1636 Desargues published the short treatise La perspective which only contains 12 pages. In this treatise, which consists of a single worked example, Desargues sets out a method for constructing a perspective image without using any point lying outside the picture field. He considers the representation in the picture plane of lines which meet at a point and also of lines which are parallel to each another. In the last paragraph of the work he considered the problem of finding the perspective image of a conic section.

Three years later, in 1639, Desargues wrote his treatise on projective geometry Brouillon project d'une atteinte aux evenemens des rencontres du cone avec un plan. One can see the influence of the work from three years earlier, but Desargues himself gives no motivation for the ideas he introduced. The first part of this treatise deals with the properties of sets of straight lines meeting at a point and ranges of points lying on a straight line. In the second part, the properties of conics are investigated in terms of properties of ranges of points on straight lines. The modern term "point at infinity" appears for the first time in this treatise and pencils of lines are introduced, although that name is not used. In this treatise Desargues shows that he had completely understood the connection between conics and perspective; in fact he treats the fact that any conic can be projected into any other conic as obvious. Although a "cone of vision" had been considered by earlier authors, the significance of this and the way that a study of conics could thus be unified had not been appreciated before.

Following Desargues' innovative work it may be surprising that the subject was not developed rapidly in the following years. That it was not may in part have been due to mathematicians failing to recognise the power in what had been put forward. On the other hand the algebraic approach to geometry put forward by Descartes at almost exactly the same time (1637) may have diverted attention from Desargues' projective methods. The first person to really carry forward Desargues' ideas was Philippe de la Hire. He had written a work on conics in 1673 before he discovered Desargues' Brouillon project. In 1679 he made a copy of Desargues' book writing:-

In the month of July of the year 1679, I first read this little book by M. Desargues, and copied it out so as to get to know it better. This was more than six years after I had published my first work on conic sections. And I do not doubt that, if I had known anything of this treatise, I should not have discovered the method that I used, for I should never have believed it possible to find any simpler procedure which was also general in application.

In fact la Hire had treated conics from a projective point of view in his 1673 treatise New method of geometry for sections of conics and cylindrical surfaces and there he had introduced the cross ratio of four points before meeting Desargues' approach. In 1685 la Hire published Conic sections which is a projective approach to conics which combines the best of the ideas from his earlier work and also those of Desargues.

Before discussing the work of Brook Taylor, with which we will end our article, let us mention that of Humphry Ditton who wrote A treatise on perspective, demonstrative and practical in 1712. This is relevant to Taylor's work since it influenced him. Ditton's book is not particularly original but he did present a geometrical approach to perspective which is carefully constructed and well written. In many ways Brook Taylor's Linear perspective: or a new method of representing justly all manners of objects which appeared three years later in 1715, is similar to Ditton's work in its quality. One notable aspect of Taylor's work was that he stated the incidence properties as axioms, making him the first to do so.

In 1719 Taylor published a much modified second edition New principles of linear perspective. The work gives the first general treatment of vanishing points. Taylor had a highly mathematical approach to the subject and, despite being an accomplished amateur artist himself, made no concessions to artists who should have found the ideas of fundamental importance to them. At times this highly condensed work is very difficult for even a mathematician to understand, and Taylor makes it clear that he is interested in the underlying principles rather than their application. The phrase "linear perspective" was invented by Taylorin this work and he defined the vanishing point of a line, not parallel to the plane of the picture, as the point where a line through the eye parallel to the given line intersects the plane of the picture. He also defined the vanishing line to a given plane, not parallel to the plane of the picture, as the intersection of the plane through the eye parallel to the given plane. As we have shown above the term vanishing point was invented long before Taylor's time, but he was one of the first to stress the mathematical importance of the vanishing point and vanishing line. The main theorem in Taylor's theory of linear perspective is that the projection of a straight line not parallel to the plane of the picture passes through its intersection and its vanishing point.

There is also the interesting inverse problem which is to find the position of the eye in order to see the picture from the viewpoint that the artist intended. Taylor was not the first to discuss this inverse problem as we saw above, one of the first to examining it had been Leonardo nearly 250 years earlier, but Taylor did make innovative contributions to the theory of such perspective problems. One could certainly consider this work as being an important step towards the theory of descriptive and projective geometry as developed by Monge, Chasles and Poncelet.

Let us end by giving examples of artists having fun with the deliberate misuse of perspective. The first is by the famous English artist William Hogarth (1697-1764) whose Perspective absurdities formed the frontispiece to J J Kirby's book Dr Brook Taylor's method of perspective made easy in both theory and practice (1754).

Books

1. K Andersen, Brook Taylor's work on linear perspective. A study of Taylor's role in the history of perspective geometry. Including facsimiles of Taylor's two books on perspective (New York, 1992).
2. J V Field, The invention of infinity : Mathematics and art in the Renaissance (Oxford, 1997).
3. M Kemp, The science of art (New Haven, 1992).
4. J-H Lambert, Essai sur la perspective (août 1752) (Coubron, 1982).

Articles

1. K Andersen, Some observations concerning mathematicians' treatment of perspective constructions in the 17th and 18th centuries, in Mathemata (Wiesbaden, 1985), 409-425.
2. K Andersen, Ancient roots of linear perspective, in From ancient omens to statistical mechanics (Copenhagen, 1987), 75-89.
3. K Andersen, Desargues' method of perspective : its mathematical content, its connection to other perspective methods and its relation to Desargues' ideas on projective geometry, Centaurus 34 (1) (1991), 44-91.
4. K Andersen, Perspective and the plan and elevation technique, in particular in the work by Piero della Francesca, in Amphora (Basel, 1992), 1-23.
5. K Andersen, Stevin's theory of perspective: the origin of a Dutch academic approach to perspective, Tractrix 2 (1990), 25-62.
6. R Bkouche, La naissance du projectif : de la perspective à la géométrie projective, in Mathématiques et philosophie de l'antiquité à l'âge classique (Paris, 1991), 239-285.
7. C D Brownson, Euclid's Optics and its compatibility with linear perspective, Arch. Hist. Exact Sci. 24 (3) (1981), 165-194.
8. M Daly Davis, Luca Pacioli, Piero della Francesca, Leonardo da Vinci : between "proportionality" and "perspective" in the Divina proportione (Italian), in Between art and science : Piero della Francesca (Italian), Arezzo-Sansepolcro, 1992 (Venice, 1996), 355-362.
9. J W Dauben, The art of Renaissance science: Galileo and perspective, a video-cassette (Providence, RI, 1991).
10. J Elkins and L da Vinci, Did Leonardo develop a theory of curvilinear perspective? Together with some remarks on the "angle" and "distance" axioms, J. Warburg Courtauld Inst. 51 (1988), 190-196.
11. J V Field, Giovanni Battista Benedetti on the mathematics of linear perspective, J. Warburg Courtauld Inst. 48 (1985), 71-99.
12. J V Field, Linear perspective and the projective geometry of Girard Desargues, Nuncius Ann. Storia Sci. 2 (2) (1987), 3-40.
13. J V Field, Perspective and the mathematicians : Alberti to Desargues, in Mathematics from manuscript to print, 1300-1600, Oxford, 1984 (New York, 1988), 236-263.
14. J V Field, Piero della Francesca and the "distance point method" of perspective construction, Nuncius Ann. Storia Sci. 10 (2) (1995), 509-530.
15. J V Field, R Lunardi and T B Settle, The perspective scheme of Masaccio's Trinity fresco, Nuncius Ann. Storia Sci. 4 (2) (1989), 31-118.
16. A Flocon, Wentzel Jamnitzer : Perspectiva corporum regularium, in Sciences of the Renaissance, Tours, 1965 (Paris, 1973), 143-151.
17. P Freguglia, De la perspective à la géométrie projective : le cas du théorème de Desargues sur les triangles homologiques, in Entre mécanique et architecture/Between mechanics and architecture (Basel, 1995), 89-100.
18. P S Jones, Brook Taylor and the mathematical theory of linear perspective, Amer. Math. Monthly 58 (1951), 597-606.
19. W R Knorr, On the principle of linear perspective in Euclid's Optics, Centaurus 34 (3) (1991), 193-210.
20. A Pucci, A mathematico-esthetic itinerary : from perspective to projective geometry, Quad. Ric. Didatt. No. 10 (2001), 63-76.
21. H Ratsimba-Rajohn and Th Bautier, Perspective conique et géométrie dans l'espace : F. Brunelleschi-aspects historiques, épistémologiques et didactiques, in Séminaires de mathématiques. Science, histoire, société, Rennes, 1984 (Rennes, 1984), 1-4.
22. A Sarounová, Geometry and painting : The origins of linear perspective (Czech), Pokroky Mat. Fyz. Astronom. 40 (3) (1995), 130-150.
23. R Tobin, Ancient perspective and Euclid's Optics, J. Warburg Courtauld Inst. 53 (1990), 14-41.
24. V Valerio, Projective knowledge and perspective of lines in the works of Ptolemy and in late Hellenistic culture (Italian), Nuncius Ann. Storia Sci. 13 (1) (1998), 265-298.
25. K Veltman, Piero della Francesca and the two methods of Renaissance perspective, in Between art and science : Piero della Francesca (Italian), Arezzo-Sansepolcro, 1992 (Venice, 1996), 407-419.