The ’Fundamental Theorem’ ofDe Prospectiva Pingendi∗

Riccardo Migliari and Marta Salvatore†

February 5, 2016

Abstract

In a mathematical discipline, the ‘fundamental theorem’ is the proposi-tion on which the deductions, the enunciations and the following demon-strations are based. There is a passage, in De Prospectiva Pingendi, whichperforms precisely this role and it is also the only passage, in the wholetreatise, which is not supplied with an illustration. This is perhaps one ofthe reasons why this passage often has been overlooked by the exegetes ofPiero della Francesca’s works: the only one who seems to grant it some sig-nificance is Rudolf Wittkower, in a piece of writing of his from 1953 [13].However, there are also other reasons that make the passage misleading: thefirst reason is connected with the mental attitude of the reader, who is in-fluenced by knowledge that has nothing to do with Piero’s world, and alsoby a way of reasoning that is very different from his. In this conditioning isplaying a significant role:

- the idea that De Prospectiva Pingendi is, solely, a treatise on perspec-tive.

- the idea that Piero makes use of the vanishing point, knowing, evenif only intuitively, the meaning of it and that he, consequently, is alsocapable to refer to the distance-point.

All this deprives the passage we are dealing with of its value in the contextof the treatise. We believe, instead, that in the passage in question is hiddenthe most important contribution of Piero to the history of representation: a

∗Published in Le teorie, le tecniche, i repertori figurativi nella prospettiva d’architettura trail ’400 e il ’700 - Dall’acquisizione alla lettura del dato, edited by Maria Teresa Bartoli andMonica Lusoli, Florence University Press, Florence 2015, entitled the ‘Fundamental Theorem ofDe Prospectiva Pingendi’ - English translation by Susanne Bredholm

†We owe to Marta Salvatore the paragraph The world as it is: the method of the Propia Forma,all the models, the figures and the bibliography

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bridge built between the Euclidean world of certainties and the projectiveworld of appearances and therefore, exactly, the ‘fundamental theorem’ ofDe Prospectiva Pingendi.

The ’fundamental theorem’ and the possible condi-tioning of a reader1

When we are about to read a book, any literary work of contemporary thought, wespontaneously place it within one of the countless classes in which the knowledgeand the arts of the modern-day man are organized: that book, therefore, maybe a literary or a scientific work, a novel or an essay, a treatise on physics orgeometry, etc. Well, with De Prospectiva Pingendi this unrefined classification isnot possible.

Piero’s treatise should be read and studied mastering many and diverse com-petencies, which involve disciplines that today are characterised by a high levelof specialisation2. De Prospectiva Pingendi is, in fact, not only a theoretical andpractical treatise on perspective: it is a document for the history of the Vulgartongue and its relationships with the Latin language; it uses in full awareness theassociated orthogonal projections three centuries before Gaspard Monge; it de-scribes the shape of a Composite capital at the same time when the architectswere formulating the theory of the architectural Orders; it is a collection of draw-ings of great value, the only drawings handwritten by Piero della Francesca thathave reached us: De Prospectiva Pingendi is therefore, in itself, a work of art.

We therefore believe that a truly in-depth and thorough reading of this workcould only be done by an interdisciplinary group of scholars, willing to listen toeach other and to help one another.

We have felt the need to make these preliminary remarks, almost obvious, alsoin order to explain our, incomplete, hodiernal reading, which relies on a single andunilateral point of view: that of the descriptive geometry. And yet in this definitiona manifest anachronism is disclosed, since the term Géométrie Descriptive hasbeen coined, as is well known, by Gaspard Monge in 1794 and it should thereforenot be used in this context. We will, nevertheless, lay all scruples aside and usethe term, in the Italian wording, because with the term ‘descriptive geometry’we mean not only the science codified by Monge, but all that pre-existed the

1The authors’ thanks go to Marisa Dalai, Carlo Maccagni, Ottavio Besomi for their commentsand precious knowledge lavished in the course of the meetings of the Committee for the NationalEdition of Piero della Francesca’s works.

2Corrado Maltese did remind us to be attentive, regarding the risk deriving from specialisedinterpretations of Piero’s work [10]

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contribution of the French mathematician, in a continuous flow that goes fromVitruvius, maybe, up until the present day, passing through Piero’s testimony.

This point of view, which explicitly places De Prospectiva Pingendi withinthe history of descriptive geometry, instead of, as is more customary, within thehistory of perspective, is legitimated by the presence, in Piero’s treatise, of twoof the methods which form, currently, the disciplinary corpus of descriptive geo-metry: the orthogonal projection method, which later was mainly attributed toMonge, and the veritable perspective. These two methods are both present inPiero’s treatise at a founding stage of their evolution, that is, in the moment of thechange from a use which is merely intuitive, to an early theoretical formulation;this defines, exactly, the boundary between what is ’method’ in the scientific senseof the word and what is empirical artifice and to a certain degree unconscious.

As for what concerns the method of the orthogonal projections, Piero, as amatter of fact, does not say a single word of comment, but his handwritten draw-ings fully overcome this incompleteness and show, without the need to make con-jectures, a profound awareness of the theoretical principles that are at the basis ofthem 3.

It is therefore worth remembering that, in the current acceptation, descriptivegeometry comprehends the representation methods and the study of the surfaces,as well as many and useful applications. With the term ‘representation method’we mean a whole series of theories and procedures which allow constructing thevirtual model of a three-dimensional object and to operate on it, as if it were a realobject. Finally, a representation method is such when it not only produces a re-presentation of the real or virtual object that it studies, but when it, as well, allowsus to perform, on this representation, geometrical operations which, at their turn,simulate real operations. For instance, in the method today known as ’Monge’smethod’, it is possible to draw a brick and to study it and measure the shapethat this brick acquires if it is intersected by an oblique plane. These operationsmay be carried out in perspective, as well as others of more common use, likethe construction of the linear shadows and the measurement and rendering of thechiaroscuro.

3On the method of orthogonal projections in De Prospectiva Pingendi have written, in parti-cular, Filippo Camerota [2] and Kirsti Andersen [1]

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De Prospectiva Pingendi, witness of the two faces ofthe world: as it is, and as it looksWe linger over this definition because Piero uses, precisely, a method of repre-sentation of which he takes for granted that his readers have thorough knowledge4

(namely the orthogonal projections); then he creates another method (the so-calledfirst manner) which serves to construct models in perspective and, finally, he pro-poses a graphical procedure (the second manner) which is used for the construc-tion of perspectives and which employs the first of the above-said methods, thatis, the orthogonal projections.

It is therefore necessary to agree on the terms and, above all, to understand thedifference between method and simple procedure.

But before that, these methods should be framed in the context of the cultureof that time, even if in its overall lines.

At that time, like today, the world presents two faces: one face shows us howit looks, and the other how it is. Here we do not want to go into the psychologicalaspects of this distinction, which perhaps is too concise; here we just want todevelop a scheme that will help us to understand Piero’s work.

To know the world as it is, there were, already in the time of Piero, sophis-ticated theories: from Euclid’s Elements to the geometry of Archimedes. Thesetheories allow measuring and describing three-dimensional shapes and to repre-sent these by means of a drawing, in two dimensions, like imprints in the sand(ichnography) or shadows projected by the Sun (orthography). To describe theworld as it looks, there is the theory which goes from Euclid’s Optics to the Quae-stiones Perspectivae written by Biagio Pelacani5.

Piero says nothing about the method he uses to represent the objects as theylook, simply because the theory is known and the applications are shared amongthe beneficiaries of his work. This does not mean, nevertheless, that that theory

4Però piglarò questo altro modo, col quale porrò parte per parte dimostrare le degradationi,nel qual modo, commo dissi nel principio del primo, è necesario intendere quello che l’omo volfare et quello sapere ponere in propia forma sopra del piano, perché commo siranno poste inpropia forma, la forza de le linee seguendo l’arte le produranno degradate, sicommo se rapre-sentano nel termine dalle linee visuali, [6] III, Incipit. (However, I will choose this other way,with which I will explain step by step the diminishing-in-size, in which way, as I said in the be-ginning of the first (book), it is necessary to understand what the man wants to do and what heknows about how to put in proper form (plan and elevation) on the plane, because depending onthe way they will be put in proper form, the force of the lines following the art will produce thediminishing-in-size of them, as if they represent themselves on the picture plane by the visuallines).

5On the geometric culture in the age of Piero, with regard to the perspectiva naturalis, see Fi-lippo Camerota [3]. On the natural perspective of Biagio Pelacani, see Graziella Federici Vescovini[8] and Valeria Sorge [12].

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and those applications do not form an integral part of the treatise; in fact theyare explicitly employed in the procedures of the third book and implicitly in themethod explained in the first and the second book.

Piero resumes the line of reasoning on the appearance and on the essence of theworld from the point where the ancients have abandoned it, and he devotes thusto perspectiva naturalis and to the ‘proper form’ of the shapes, the first elevenpropositions of his treatise (see Appendix). Piero expands then, autonomously,the aforesaid reasoning in the passage included between the propositions 11 and12, which we here call ‘fundamental theorem’6 and, doing so, he founds the per-spectiva artificialis and he builds a bridge between the essence and the appearanceof the objects, between the ancient world and the modern world.

Since Euclid’s propositions of the optics refer to the reduction of the sizes dueto the distance from the onlooker, on this aspect of the appearance of the objects,in particular, are founded the theoretical developments formulated by Piero and,particularly, the ‘fundamental theorem’ of which we will speak later.

The world as it is: the method of the Propia FormaFirst, though, it is worth briefly to consider the theories and the applications whichrefer to the world as it is.

Well, the method, which enables to describe the objects as they are, is the onethat today is known as ‘Monge’s method’. Piero uses this method in a slightlydifferent form, paradoxically more general, even though poorer.

This representation consists of two associated drawings: the ichnography ofVitruvian memory, which Piero calls the figura della larghezza (image of thewidth)7 and the orthography, which he calls the figura della altezza (image of theheight)8, with evident reference to the measurement operations that can be doneon it, seeing that the plan preserves the propia forma (proper form) of the hori-zontally positioned geometric entities (lines and plane shapes), and the elevation

6The term is closed in inverted commas since it is improperly used: Piero, in fact, does not em-ploy a logical-deductive exposition and he does therefore not proceed by applying theorems. Hisculture and his way of dealing with problems are still those of the abacus school. His enunciationsconcern for the most part particular cases, his demonstrations are empirical. Nonetheless, in thepassage in question, Piero faces the problem of the diminishing in size of the apparent quantitiesfrom a mathematical point of view and he gives, as we will see, a general solution.

7Hora per fare la figura de la larghezza in propia forma, piglia la quantità de la pianella de labasa, ch’è .GH., et fà una superficie quadrata ..., [6] III, 6. (Now, to create the shape of the widthin proper form (plan and elevation), take the measure of the plinth at the base, namely GH, andcreate a square surface).

8... et ai compiuta la figura de l’altezza in propia forma, [6] III, 6. (And you have made theshape of the height in proper form).

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preserves the true shape of the vertical segments9.Through this first representation Piero is able to draw the representation of the

same object in a generic position, in which the faces and the edges are not ‘inproper form’ anymore.

Let us consider, for instance, proposition V of the third book: Il cubo datoposante sopra ad uno suo angulo, et che nisuno suo lato sia equidistante al ter-mine posto, proportionalmente degradare (Given the cube placed on one of itsvertices, and that none of its sides is parallel to a given picture plane, will diminishproportionally); this proposition is dedicated to the perspective of the hexahedron.

The polyhedron is at first represented in a plan and elevation form, resting withone face on the ichnographic plane (horizontal), with its vertical edges paralleland the vertical faces oblique with respect to the orthographic plane (vertical) andtherefore in a particular position. To the solid is then applied a translation, fromleft to right and from above downwards, and a rotation of about eighty degrees,which pivots on a vertex and moves the others on planes that are parallel to theplane of the orthography (Fig. 1).

The procedure, irreproachable, documents the capability to use a representa-tion to simulate operations which involve the three dimensions, as it could be donein the real space, and it thus shows the effectiveness of the method. This techniquebelongs to the motions that nowadays are known as ‘change of projection planes’,since the motion of an object with respect to projection planes is equivalent to themotion of the projection plane with respect to the object, and that allows solv-ing many problems related to plane sections and the true shapes of the geometricfigures.

Gaspard Monge widely makes use of the sliding motions in his Leçons [8],to solve the most complex intersection problems and for other reasons, bestowing‘his’ geometry with that visual and dynamic characteristic which gives it most ofits charm.

We would now like to consider another aspect of the method so far discussed,which we from now on will call, to pay homage to Piero, ‘method of the propiaforma’, since we cannot call it ‘method of Monge’ for obvious reasons.

This aspect is the one that concerns association between the two images of thewidth and the height, or better the plan and elevation. These two images, in fact,always have to be associated with each other, since in the first one we read theposition of the point with respect to the horizontal plane (the width), whereas inthe other we read and we measure the level of the point (the height); in this wayany point P, of which is given the first and the second image, can be replaced inthe space which is situated above the drawing of the width.

9. . . è necesario intendere quello che l’omo vuol fare et quello sapere ponere in propia formasopra del piano [6] III, Incipit.

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Figure 1: Representation of the hexahedron in generic position, from the Palatinecode in Parma. The solid makes two rotations: one about the vertical axis, thesecond turning about one of the vertices. The slight translation downwards is anexpedient to avoid that part of the shape falls out of the upper edge of the page.

Well then, even if Piero does not say a single word about these issues, it islegitimate to say that he is perfectly acquainted with the relation that the three-dimensional model of the subject of his studies establishes with the drawings ofthe proper form and with the perspective, that is to say that he is fully aware of thatspatial configuration which we nowadays, with our means, easily can represent ina virtual space.

As a matter of fact, each time he can, Piero clearly expresses this relation,uniting the images of the same objective point by means of lines of reference (theones we nowadays call reference-lines)10.

This is manifest in the drawings related to the propositions 5 and 8 of the thirdbook, the first dedicated to the perspective of the hexahedron, which we mentioned

10See Monge [11], pag. 14: si on a la projection horizontale d’un point la projection de ce mêmepoint sur le plan vertical supposé abattu sera dans la droite, menée par la projection horizontaleperpendiculairement à l’intersection LM des deux plans de projection, et réciproquement. Cerésultat est d’un usage très-fréquent dans la pratique (If we have a horizontal projection of a point,the projection of this same point on the vertical plane presuming that the rabattement is performedabout a straight line, draw through the horizontal projection perpendicularly to the intersectionLM of the two projection planes, and reciprocally. This result is very often used in practice).

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above, the second dedicated to the perspective of a human head, seen from an eye-level point of view and from another, that is, seen from below upwards.

There are, nevertheless, cases in which Piero cannot use this artifice. In therepresentation of the Italic capital, for instance, these lines are never present, evenwhen the two images of the width and the height are situated on the same page ofthe manuscript. Why?

We believe that this happens when the image of the width and the one of theheight have such a complexity that they cannot be contained on a single sheet ofpaper, and when, if necessary, the drawing is carried out on two distinct sheetswhich are as big as possible, in order to reduce the graphic errors and to avoidmisinterpretations.

In these cases, the drawing that the manuscript shows us is just an allusion, in-complete and simplified, to another drawing, the original of which, unfortunately,is lost; but this original drawing can be reconstructed by the reader thanks to thevery accurate description that the text gives of it, guiding him step by step.

And, on the other hand, in architectural drawing, to this day, this is exactlywhat is in use: we separate the drawings of the plan and the elevation, placingthem on two distinct sheets of paper, which thus prove to be of easier executionand of more prompt consultation, even if this, prior to the advent of digital tech-nologies, leads to errors and misinterpretations.

We should therefore not be surprised about the use Piero is making of therepresentation in propia forma to generate a perspective, seeing that he mastersthe method, but we have to be mindful of the fact that that perspective, the onehe teaches in the third book, is generated, in a representation in propia forma, asa result of numerous intersection operations and, therefore, it is not a method initself, but rather the result of an iterative procedure.

To fully appreciate this difference, it is worth to observe that in the mannerperformed in the first book and, more plainly, in the second book, the perspectiveis constructed working directly in the perspective space.

The solids that are represented arise from an imprint on the ground that alreadyis a perspective and they elevate in a space which is that of the world as it looks,thus in an anisotropic space, in which the heights depend on the distance fromthe onlooker, and with the distance the objects decrease in size until they almostdisappear.

In other words, modern words, in this which really is a method, three-dimensionalshapes are constructed in the projective space, that is, in the space of the world asit looks.

In the third book, instead, the prospective originates in the isotropic space ofthe propia forma, which is ruled by the Euclidean laws, and the three-dimensionalshapes are constructed in the space of the world as it is.

Thus the De Prospectiva Pingendi is not ‘only’ a treatise on perspective, it is

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a treatise which interrelates the space as it is with the space as it looks, and themethod of the propia forma with the method of perspective.

The De Prospectiva Pingendi is, therefore, the earliest evidence of the scienceof representation that nowadays is known as ‘descriptive geometry’, in the broadsense we outlined before. And we should also specify that, with regard to themethods, the perspective of Piero’s earliest manner is the most general case, inwhich are included the methods, considered as specialisations of it, which makeuse of the orthogonal projections. And therefore, looking at it more closely, de-scriptive geometry is a particular case of perspective [9].

As for the relationships between the method of perspective and the method ofthe orthogonal projections, they restrict themselves to the correspondences amongpoints of the finite space, which Piero interweaves in the third book, and such theyremained until, with Kepler [9] and Desargues [10], geometry was enriched withthe concept of the point and the straight line at infinity: then, and only then,perspective gave a meaning to the point at infinity. The consequences of thisachievement were immense:

- the general shape of a sheaf of lines that are incident in a point, will admit,as a particular case, a sheaf of parallel lines that have they support in a point atinfinity (but d’une mesme ordonnance des droictes [7]) and analogously for thebundle of planes;

- perspective will become a tool which is capable to treat infinity in finiteterms;

- the Euclidean space and the projective space will merge into one, since thespace of the objects as they are, is only a particular case of the space of the objectsas they look.

The world as it looks: the method of ProspectivaThe moment has come to deal with the method of perspective that was, for thefirst time, theorised and experimented by Piero della Francesca in De ProspectivaPingendi and, to fully understand the value of this theory, its principles, its revo-lutionary impact and, at the same time, its limits; we will need to briefly take intoconsideration the contents of the first book, in which these concepts are developedand applied to the perspective of the plane shapes.

The first book comprehend thirty propositions, of which the first eleven canbe considered propaedeutic (see Appendix) and the others, from the twelfth on,essential of the first manner.

The eleven propaedeutic propositions play the role that in the current treat-ment of perspective (understood as a method of representation) is performed bythe operations of projection and section and by the concepts that lead to the pro-

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jective enlargement of the Euclidean space. This role consists in justifying theconvergence, in the illusory space, of the images of the straight lines which in thereal space are parallel.

In retrospect, perspective is all here, all included in the point that Guidubaldodel Monte, one hundred years later, will call Punctum Concursus and Brook Tay-lor, another century later, Vanishing Point.

In the present day conception, which derives from the slow evolution of theidea of geometric infinity11, the vanishing point justifies and measures the per-spective view. This means that, to construct the correct decrease in size of iden-tical quantities positioned at different distances, we need particular points (called‘measurement-points’ and, in special cases, ‘distance-points’).

In the conception of De Prospectiva Pingendi, instead, it is exactly the oppo-site: it is the perspective view which suggests the vanishing point, or rather ‘theeye’, namely puncto A12.

What I want to say is that Piero, being unable to justify geometrically the

11Ugo Cassina [4], in a memorandum perhaps a little old-fashioned but still very useful, sum-marises the fundamental stages of the evolution of the conception of geometric infinity.

- The first move towards the current conception was made by Guidubaldo del Monte in 1600,when he identifies in the punctum concursus the point intersection with the picture planeof the straight line that passes through the eye and that is parallel to the lines (horizontal)considered.

- The second step is attributed to Johan Kepler in 1604. Kepler identifies in the parabola afocus point at an infinity-distance (both in one direction of the axis and in the other); heformulated, thus, the idea that parallel straight lines in a plane are passing through a pointthat is common to them ‘at infinity’.

- The third step is performed by Girard Desargues in 1639, when he formulated the defi-nitions of the but d’une ordonnance des droictes (improper point) and the ‘but d’une or-donnance des plans’ (improper straight line), that is, of our conception of direction andlying-position, which are the entities that perspective shows in the vanishing point of asheaf of parallel straight lines and in the vanishing line of a bundle of parallel planes.

It is not necessary to go any deeper into the progression of this idea, since it is enough, I believe, torecall what we already have said, to acknowledge that in Piero’s point A there is something differ-ent from the vanishing point, insomuch that it may well be appropriate not to use this expression,to minimize the risk of any misunderstandings.

12Piero calls ‘eye’, without distinction, what today are known as the projection centre and theprincipal point, or vanishing point of the lines perpendicular to the picture plane, that is, two well-distinct points: the first is situated in the space and the second on the picture plane. This usageis absolutely natural, in fact, if we place ourselves before a mirror, looking with only one eye,the principal point coincides exactly with the pupil of the eye reflected in the mirror. And if wemeasure the height of the face, for instance, from the hair to the chin, we will find out that itmeasures one-half of the real dimensions, always and in any case, whatever distance and standingposition we assume with respect to the mirror. These empirical observations are very likely at thebasis of the theoretical developments that will ensue.

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vanishing point, of which he has only a partial evidence in the visual convergenceof the lines that are perpendicular to the picture plane, he structures his reasoningin this way: at first he studies the diminishing in size of the quantities and triesto express it using a mathematical law, then he applies this law to the side of thesquare, parallel to the picture plane, which is situated farther from the observer,finally he extends the two straight lines which represent the sides of the samesquare that are perpendicular to the picture plane; he thus obtain the point in whichthese straight lines are incident as an empirical datum, validated by the recurringresults of the aforesaid construction13.

The co-existence, in the graphic constructions, of different representations ofthe same perspective machine, as we will see in the thirteenth proposition, sug-gests that that point A in which, as we have already said, converge the perspectivesof the straight lines perpendicular to the picture plane, has some relationship withthe eye of the onlooker, or rather with the projection centre; but nothing more.

In fact, the identification of that point, as an intersection with the picture planeof the projecting line that is parallel to the observed straight lines, is an achieve-ment that arrives, as we said before, with Guidubaldo del Monte. And if Piero hadhad knowledge of it, he would not have had any difficulties in identifying othervanishing points, like that of the straight lines inclined at 45 degrees (that is, thedistance point) and like that of any direction, support of a sheaf of parallel lines,in a generic position.

This difference with the present-time conception of the vanishing point is notinsignificant, since it is a proof of an unsolved difficulty: the paradox of a conver-gence that seems to contradict the twenty-third definition of the first book of theElements and, implicitly, the fifth postulate, in the case the perspective representsas being incident straight lines that should not have any point in common.

Nevertheless, after having consolidated the aforesaid experimental result, Pieroembraces it and uses it to create an abbreviated construction of the diminishedsquare which is, very simply, a section of the triangle that has vertex in punctoA; all that in a way which is not dissimilar from the one still practised nowadays,paving thus the way to the achievements of Guidubaldo, at first, and Brook Taylorlater.

Coherently with the general difficulty mentioned in the foreword, it is not easyto distinguish our interpretation of the thought of Piero della Francesca from hisoriginal conception, and the vanishing point is exactly the confine that we willtake a closer look at.

To us, the vanishing point is the image of a point situated very far away or, toput it in geometrically exact terms, the perspective representation of the direction

13Andrès de Mesa [5], already in 1989, draws the attention of the scholars of the earliest signs ofperspective against the risks of a too easy, as well as misleading, reading of the ‘vanishing point’.

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of a sheaf of parallel lines.To Piero, the vanishing point is simply the point of the picture plane in which

the perspectives of the straight lines that are perpendicular to the picture planeseem to converge.

Piero does not attribute any other significance to this point, nor he explains ortries to explain why this happens; he only shows, with Euclid, that it is so, and hedoes it restricted to a particular class of straight lines and not for each class, thatis, for the straight lines that have a generic direction.

But, let us get back to the propositions of the first Book.The first four do only consider, geometrically, the various illusory effects of

a perspective: quantities which, although being dissimilar, appear to be identical,due to the distance (proposition 2), visual angles that reveal differences of sizeand distance (3 and 4). In the successive propositions, some (6 and 7) continueto describe the illusory effects, whereas others (from 5 to 11) define the relationsthat derive from divisions carried out on the proper form of the shapes.

From the eleventh proposition on, explicitly devoted to the diminishing of theapparent sizes, and thus to the perspective views, Piero defines, for the first time,and in mathematical terms, the principal distance as being a module for the dimini-shing of the apparent sizes; this passage can therefore, rightfully, be considered asthe ‘fundamental theorem’ of the first manner for creating a perspective.

What, with a definition which is widely accepted nowadays, is called princi-pal distance or, also, focal distance, is the distance which separates the projectioncentre from the picture plane, or rather the ‘eye’ from the termine dove se mecte lecose degradate (the picture plane where the diminished image is positioned). Theperspective view can be described mathematically as a relationship between thisdistance and the distance of the objects seen from the viewpoint: ... perché quandodico proportionalmente ch’èproportione intendo, perché le proportioni sono inu-merabili, et questa nonnè dupla commo è.2. et .4. e.8., et nonnèsexquialteracommo .4., .6., .9., nèsexquitertia commo .9., .12., .16., nè tripla, nèquadrapla,ma dico essere proportione degradata, non commo .4., .8., .12., .15., nè commo.6., .9., .11., .12., ma è secondo la distantia da l’ochio al termine dove se mecte lecose degradate et la distantia dal termine a la cosa veduta.(Della Francesca1942:73-74) ( ... because when I say proportionally it is proportion I mean, because theproportions are innumerable, and this is not double like 2 and 4 and 8 are, and notsesquialtera like 4, 6, 9, nor sesquitertia like 9, 12, 16, nor triple, nor quadruple,but I say it is proportion diminished-in-size, not like 4, 8, 12, 15, nor like 6, 9, 11,12, but according to the distance from the eye to the picture plane where the di-minished image is positioned and the distance from the picture plane to the imagethat is seen).

This point deserves, therefore, further consideration. Given an observer, thatis, a point of view, and some objects separated by intervals that indeed look equal

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to the observer who is standing in front of them, the perspective changes, if the dis-tance of the observer from the aforesaid object changes, and precisely: if the on-looker is nearer, the apparent space-intervals broaden and the difference betweenthose nearer and bigger, and those more distant and smaller, is more evident. Andthis is how the perspectiva naturalis behaves (Fig. 2).

Figure 2: Variations of the perspective in function of the distance of the onlookerfrom the observed objects.

If we introduce the picture plane and the projection operation that has its cen-tre in the eye of the observer and produces the image on the picture plane, thingsbecome more complicated and precisely: if we translate the picture plane movingit closer to the onlooker, the perspective does not change its shape, it simply be-comes smaller; if we translate the picture plane moving it further away from theonlooker, the perspective does not change its shape, it becomes larger. When oneof the objects touches the picture plane, the portion that the object has in commonwith it is in full-size and does not undergo any perspective deformation (Fig. 3).

As a consequence, two factors intervene in perspectiva artificialis: the first isthe distance of the observer from the objects that give to the perspective its ‘shape’.The second is the distance of the observer from the picture plane; this distancegives to the perspective its ‘scale’, that is, the ratio of reduction or enlargement of

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Figure 3: Variations of the perspective in function of the translation of the pictureplane, given a fixed position of the onlooker with respect to the observed objects.

the perspective image14.Piero’s conquest lies in the fact that he succeeded in explaining, with one

single and simple law, the interaction between the two aforesaid factors, by meansof a sequence of ratios in which the denominator of the first considered elementrepresents the distance of the onlooker from the picture plane and the sequence initself the diminishing in size of the quantities perceived by that onlooker.

Now, in order to better understand this important moment in the history ofperspective, it is necessary to get to the heart of Piero’s reasoning.

In the passage at issue, which is not illustrated, of the manuscript conserved

14The comparison with photography is inevitable, but it introduces a further complication be-cause in photography the picture plane has finite and fixed dimensions, whereas in perspective itis indefinitely extended. Therefore, if we keep the position of the camera-body still and we usea wide-angle objective (in which the picture plane is near the projection centre) the perspectiveembraces a wide field angle; if the camera-body is equipped with a telephoto lens, the perspectiveembraces a restricted field angle; but the perspective image, that is, the shape of the perspective,does not change, as it has been shown many times before, empirically, too. The impression thatbetween the two images there is a difference in the compression of the space, because the firstshows an expanded space and the second a compressed space, is only due to the fact that the firstimage highlights also the objects and the part of space that is nearest to the onlooker, whereas thesecond image emphasizes the objects that are farther away.

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in Parma, Piero says:Cioè così: sono quatro linee equidistante, et l’una da l’altra è uno braccio, et

sono lunghe uno braccio, et sono infra do linee paralelle, et da la prima linea cheè termine a l’ochio è quatro braccia; dico la seconda a la prima essere sexqui-quarta, et la terza a la seconda nel termine è sexquiquinta, et la quarta a la terzanel termine è sexquisexta’. (That is, like this: there are four parallel lines, and thedistance between the one and the other is an ell, and they are an ell in length, andthey are between each other parallel lines, and from the first line which is the pic-ture plane to the eye is four ells; I say that the second to the first is sesquiquarta,and the third to the second on the picture plane is sesquiquinta, and the fourth tothe third on the picture plane is sesquisexta).

First of all it is necessary to explain the exact meaning of the terms sesqui-quarta, sesquiquinta and sesquisexta. They mean, in the proportions, a fraction inwhich the numerator is equal to the denominator plus one, namely:

(x + 1)/x ;

thus, sesquiquarta proportion, where x = 4, means (4 + 1)/4, and thereforesimply 5/4; and thus sesquiquinta means 6/5, sesquisexta 7/6 and so on.

Piero’s reasoning can be read as follows (Fig. 4).The eye, which is positioned in point A, looks at the line BC conventionally

situated on the picture plane (the termine in Piero’s language), and sees as wellthe lines DE, FG, HI that are spaced out at equal intervals.

The primary distance is AB and it measures 4 ells, while BD, DF, FH areintervals of one ell each. The quantities BC, DE, FG, HI etc. measure each anell, too, like the intervals which separate them.

Given that BC is not subject to any perspective deformation, since it is situatedon the picture plane, we wonder: which is the perspective deformation of DE?

And then: in which ratio are, among them, the quantities of the foreshortenedviews generated by the perspective?

The ratio between BE’ and BC, where BE’ is the perspective of DE, due tosimilarities of triangles, will therefore be:

BE’AB = DE

AD

but

BC = DE

and thus

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Figure 4: Graphic transcription of Piero’s reasoning, with a sequence of horizontalline segments (above) and vertical (under).

BE’AB = BC

AD

therefore

BE’ =AB·BC

AD

and since AB = 4, BC = 1 and AD = 5 the ratio is 4/5, which means that BE’is 4/5 of BC and BC is 5/4 of BE’.

The text nevertheless says: dico la seconda a la prima essere sexquiquarta(I say that the second to the first is sesquiquarta), where secunda should be DE,which, seen from A, is referable to BE’, and prima should be BC; the right ex-

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pression ought, therefore, to be the first one, which, though, is worth 4/5 and not5/4.

Summarising: BE’ is smaller than BC and, precisely, the 4/5 of BC, but Pieroseems to say that BE’ is to BC as 5 is to 4.

An interpretative issue therefore arises: why does Piero say ‘the second to thefirst’ (ratio equal to 4/5) and not the opposite, namely ‘the first to the second’(ratio equal to 5/4)?

Can we suppose that he minimizes the importance of these mathematical con-ventions? After all, when he names a line segment, he does not distinguish thedirection (AB and BA is the same thing): with a ratio he could do likewise. Thefact remains, however, that this ambiguity may mislead15.

Let us now consider the other quantities, referring them all to the first, BC.Leaving out the proportions it emerges that, if we ask ourselves how much smallerthe perspective deformation is than the whole quantity, we will obtain the se-quence:

(4/5), (4/6), (4/7)

which is read as: the second is 4/5 of the first, the third is 4/6 of the first ... andso on.

If we do the opposite, asking ourselves how much bigger the whole quantityis than perspective deformation, the sequence will be:

(5/4), (6/4), (7/4)

and this sequence is read as: the first is 5/4 of the second, the first is 6/4 of thethird ... and so on.

Let us now come to the comparison which concerns Piero: it is not that ofthe single lines shortened with respect to the entire line segment, but that of theshortened lines among them; it is as if he would have liked to follow, step bystep, the diminishing-in-size, ‘entering’ into the perspective. We will try, later, tounderstand why.

If we accept the first sequence and we ask ourselves, thus, how much smalleris the perspective foreshortening of a line than the one which precedes it, the

15Into this trap falls, for instance, Alessandra Sorci in her work La forza delle linee [11], page86. This is the interpretation of the passage: Poiché la seconda linea della serie è lontana cinquebraccia dall’occhio, la sua proiezione sulla prima sarà eguale ai 5/4 della lunghezza effettiva ...(Since the second line of the series is distant five ells from the eye, its projection on the first willbe equal to 5/4 of the real length). But, the second ‘line’ is located farther than the first and it cantherefore not appear to be bigger; on the contrary it will look smaller and, precisely, 4/5 of thefirst.

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diminishing-in-size becomes:

(4/5) : 1 = (4/5)

(4/6) : (4/5) = (5/6)

(4/7) : (4/6) = (6/7)

and the sequence is:

(4/5), (5/6), (6/7)

which is read as: the second is 4/5 of the first; the third is 5/6 of the second;the fourth is 6/7 of the third.

If, instead, we accept the second sequence and we ask ourselves how muchthe foreshortened view of a quantity that is nearer to the onlooker is bigger thanthe one which follows, the diminishing-in-size will be:

1: (4/5) = (5/4)

(4/5) : (4/6) = (6/5)

(4/6) : (4/7) = (7/6)

and the sequence is:

(5/4), (6/5), (7/6)

which is read as: the first is 5/4 of the second; the second is 6/5 of the third;the third is 7/6 of the fourth.

At this point Piero seems to foresee the difficulties of the reader, either due tothe Latin terms sesquiquarta etc. which only a few can immediately understand,say because of the difficulty to follow an abstract reasoning; he therefore expressesthe aforesaid ratios using the example of a numerical sequence, in the ways of theabacus school: 105, 84, 70, 60.

If we apply the aforesaid considerations to this sequence, we obtain, in thefirst case:

84/105 = 0.8 ; (4/5 = 0.,8)

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70/84 = 0.83 ; (5/6 = 0.83 . . . )

60/70 = 0.85 ; (6/7 = 0.85 . . . )

and, in the second:

105/84 = 1.25 ; (5/4 = 1.25)

84/70 = 1.2 ; (6/5 = 1.2 . . . )

70/60 = 1.17 ; (7/6 = 1.17 . . . )

the correspondence is, obviously, perfect.Thus the note written by Giusta Nicco Fasola [12] in which she says that La

legge fisicamente esatta della degradazione fu trovata solo da Leonardo, secondola progressione 1/2, 1/3, 1/4, 1/5 (The correct physical law of the diminishing-in-size has been found only by Leonardo on the basis of the progression 1/2, 1/3, 1/4,1/5) and that la progressione di Piero dà una diminuzione minore specialmenteper le prime grandezze (Piero’s progression gives a smaller diminishing-in-size,particularly for the first quantities) is devoid of any sense.

As a matter of fact, as shown above, to achieve the ratios described by Pierowe have, necessarily, to pass through the formulation that Nicco Fasola attributesto Leonardo. And, at any rate, Piero’s calculations are exact.

There is one more question to ask, perhaps more important than the one whichconcerns the order of the comparisons among the quantities, mentioned above.The question is this: why does Piero study the relations that intercede among theperspective deformations of the observed quantities instead of expressing, moresimply, the relation among the line in proper form and the length of each of theforeshortened lines?

We believe because of two reasons:

- the first, to verify if there exists a constant law of the diminishing in sizeof the apparent quantities, as Wittkower recognises [1] as well, but withoutgoing into the merits of this issue;

- the second, to express the aforesaid law: invariant to the varying of thedistance of the onlooker from the observed quantities and from the pictureplane; this is really the heart of the question.

Now, Piero is convinced that, by varying the distance of the onlooker fromthe picture plane and from the quantities that are associated to it, the perspectivedeformations of the four straight line segments that he considers in his example

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do vary, too, and namely the relations between the whole quantities and the dimi-nished quantities; and Piero is right. Word for word: Dunqua, mutando termine,se muta proportione (Thus, changing the picture plane, the proportion changes).

As a matter of fact, if the observer positions himself, for instance, at a distanceof six ells from the picture plane where the segment BC is, without changing any-thing else, the sequence

(5/4), (6/4), (7/4)

becomes:

(7/6), (8/6), (9/6).

And yet, the sequence of the relations among the perspective deformationswhich originates from these, becomes:

(7/6), (8/7), (9/8)

relations that still belong to the sequence previously calculated, appropriatelyexpanded:

an = (x + 1)/x = . . . (5/4), (6/5), (7/6), (8/7), (9/8) . . .

The ratio (7/6), which at first was related to the couple third/fourth quantity,remains now unchanged as a relation between the first and the second quantity.

The distance between the observer and these quantities (third and fourth in thefirst case, first and second in the second case) does not vary, and the angle that thesame quantities subtend in the eye does not change either.

Therefore, Piero found a constant of the law governing the diminishing-in-sizeand it is exactly in the way the relations are constructed, following the law

an = (x + 1)/x.

Here it becomes even easier to understand the reason why Piero chooses tostudy the relations among the foreshortened views, instead of the relations be-tween the perspective deformations with respect to the true quantity: because,doing so, one free oneself from the principal distance, that is, the picture plane;and because a law that also can be applied to the perspectiva naturalis, as well asto the artificialis, is expressed.

The theory enunciated here has an easy and direct application, which Pieronevertheless did not develop, preferring the graphic constructions that will ensue.

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It is not correct to include in the analysis of a text a consideration which goesbeyond its simple postulate, but in this case we feel like making an exception,because the study of this application permits us to better appreciate the meaningof the proposition.

Let us suppose that we are drawing the perspective of a square in a way that thefront side, nearest to us, is distant four ells from the observer; this side, BC, willbe in length 5/4 of the perspective of the successive side DE, and to construct theperspective of DE it is enough to divide BC into five equal parts and to considerfour of these: this will be the length of the perspective of DE.

If we now would like to construct the perspective of another line segment FG,of the same length, positioned at the same distance from DE, we only have todivide DE into six parts and to construct the perspective of FG in a way that itslength is equal to five of these parts . . . and so on.

If, instead, the first of the sides that we took into consideration is distant fiveells from the observer, and not four like in the preceding example, to obtain theperspective deformation of the side that follows, we have to divide it into sixparts and to take five of these. In other words, this formula is able to generate aperspective relating the reciprocal distances of the quantities that are observed tothe distance of who is observing them (Fig. 5).

Figure 5: - Application of the ‘fundamental theorem’ to the construction of theperspective of a series of squares.

Finally, the comparison between the sequence suggested by Piero and the oneproposed by Leonardo is enlightening (Fig. 6).

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Figure 6: The first sequence, at the top, shows the ratios described by Piero; theseare valid regardless of the position of the picture plane. The second and the third,show the ratios found by Leonardo in two different positions of the picture plane.

So, Piero proposes the sequence an = (x + 1)/x instead of the sequence sug-gested by Leonardo which is much simpler, namely, an = 1/(x + 1).

In the sequence written by Piero, the position of the picture plane is not signifi-cant: even if varied, the sequence remains the same and the denominator indicatesthe principal distance (of the onlooker from the picture plane):

. . . 2/1, 3/2, 4/3, 5/4, 6/5, 7/6, 8/7, 9/8, 10/9 . . .

For instance, if the observer is positioned at a distance of one ell from thepainting (position 1 in the Figure), the first side of the square which is diminishedin size is twice the following one; the second is 3/2 of the third and so on . . .

If, instead, the observer is positioned at a distance of four ells from the pictureplane (position 4), the first quantity is 5/4 etc. as already said several times.

But, in the sequence written by Leonardo the position of the picture plane isfixed: modifying its position, the sequence changes and it is necessary to rewriteit, dissimilar.

If, for instance, in Leonardo’s sequence, we relocate the picture plane movingit from position 1 to position 4, the values that are drawn from the similarity ofthe triangles become:

. . . 1, 4/5, 4/6, 4/7, 4/8, 4/9 , 4/10 . . .

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Piero’s sequence expresses the contraction of the projective space (as we saytoday) and it is independent from the projection operation; it is therefore the lawof the perspectiva naturalis and it is a general law.

Whereas, Leonardo’s sequence expresses the law of the diminishing-in-sizegenerated from a projection and section operation on the picture plane and it is,therefore, the law of the perspectiva artificialis, in the particular case in which thepicture plane is situated in position 1.

Piero’s law, nevertheless, operates in perspective as properly as Leonardo’sdoes, and it therefore welds the two worlds together, the ancient one of perspectivanaturalis with the modern one of perspectiva artificialis; and it also welds togetherthe two spaces: the Euclidean space and the perspective space (or projective).

All this reveals, once more, that, in the perspective of the very first attempts,before Guidubaldo del Monte, the vanishing point and, as a consequence, thedistance point, are of minor importance. What really matters is the geometriccontrol of the perspective deformation, namely of the diminishing in size of theapparent quantities.

The convergence of the images of the straight lines that are perpendicular tothe picture plane (and among them therefore parallel) is an empirical datum, ma-nifest, but otherwise inexplicable. The perspective diminishing-in-size, on thecontrary, is a datum which from being empirical becomes theoretical and general,through the enunciation of a law. If this law, in the end, in its innumerable possi-ble applications cannot be con numeri dimostrare (proved by means of numbers)is of little importance, since the geometric relations that it expresses can be ob-tained with simple graphic constructions which construct the objects directly inthe anisotropic space of the perspective (first manner) or which simulate the inter-section of the visual pyramid with the picture plane in the isotropic space of theorthographic representation.

ConclusionsWe believe that the first lesson that can be drawn from De Prospectiva Pingendi isthat of the inseparable coherence of the disciplinary components which form thecontent of it: the art of drawing, the fundamentals of geometry and mathematics,the physical and philosophical concept of space, the language as an expression ofa scientific thought but also in its literary acceptation.

And even more, in this overall picture of harmonious complexity, the coher-ence of the representation methods has to be perceived, as well as the birth eventof that science which afterwards will be known with the late eighteenth-centuryterm coined by Monge.

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Therefore studying, in the treatise, only the aspects concerning perspectivemay be misleading, just as it would be misleading to listen to the voice of a singleinstrument playing in an orchestra.

In the preceding pages we, therefore, wanted to get an overall view of thecontents of De Prospectiva Pingendi and, in particular, of its first book.

First of all, we identified in Piero’s work a bridge built between the essenceand the appearance of the objects and, thus, between the Euclidean geometry andperspective, between the ancient world and the modern age which starts, precisely,conventionally, with the year that Piero died.

We then studied the geometric propositions described in the first book, pay-ing particular attention to the one that distinguishes the theorems of perspectivanaturalis from the theorems of perspectiva artificialis. And seen that this propo-sition, unlike all the others, is orphaned of an illustration and a name, we calledit ‘fundamental theorem’ of Piero della Francesca’s perspective, acknowledgingits rightful role, namely: that of a verity capable of giving irreproachable scien-tific validity to all the propositions, the assumptions and the verifications that willfollow.

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Appendix.The first eleven proposition of De Prospectiva Pingendi

1. Omne quantità se rapresenta socto angolo nell’ochio (Each quantity is re-presented within an angle in the eye), which implicitly resumes the first twodefinitions of the Optics on the geometric model of the vision. Euclid sug-gested a three-dimensional model of vision, assimilable to a cone that hasthe eye as its centre, as its generatrices the ‘visual beams’, that is, the pro-jecting straight lines, and as its ‘basis’ the limits, nowadays called apparentcontour, of what is seen. The second part of Piero’s proposition alludes im-plicitly to an analogous spatial model, while the example proposed in thethird part, and the related image, represent the model in its plane shape.

2. Tucte le base vedute socto uno medessimo angolo, benché le sieno diver-samente poste, s’apresentano a l’ochio equali (All the ‘bases’ (lines in thiscase) seen within the same angle, even if placed in different positions, ap-pear, to the eye, to be identical), which implicitly resumes part of the con-tents of proposition 16 of Optics, in which is shown the inequality betweentwo line segments owing to that property.

3. Se più base fossero a l’angolo loro oposte orthogonalmente, quella ches’apresenta socto magiore angolo o ella è magiore o ella è a l’angolo piùpropinqua (If some of the ‘bases’ were positioned before the eye, the onewhich subtend the widest angle is either the biggest in size or the nearest),which explicitly resumes proportion 24 of the First book of the Elements,in which is shown how two triangles that have two equal sides and unequalthe angle between them, also have unequal bases.

4. Se da un punto se partissero linee sopra a do base equali et una fussepiù propinqua che l’altra, la più propinqua farà magiore angolo nel dictopuncto (If from one point originate lines over two equal ‘bases’ and onewere nearer than the other, the nearest will have a wider angle in said point),which implicitly resumes the contents of proposition 5 of Optics, in whichinstead is shown how equal quantities that are positioned at different inter-vals appear to be unequal: bigger the nearest one, smaller the farther one,due to the width of the angle subtended by the visual beams drawn to theirextremities.

5. Se da um punto se partisero do linee recte sopra ad una basa et sopra diquelle venisse un’altra linea equidistante a la basa e fusse a quella equale,deviderà sé e quelle due in una medessima proportione, et quella parte dela line[a] sopravenuta, che infra le duo linee è interchiusa, sirà colla basa

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in quella proportione che le linee intere colle loro parti terminate al puncto(If from a point originate two straight lines over a ‘basis’ and over thesewere placed another line that is parallel to the ‘basis’ and identical to it,this will separate itself and the two others into an identical proportion, andthe part of the supervened line, which is located between the two lines, willit be with the ‘basis’ in the proportion that the intern lines with their partsterminate in the point), which implicitly resumes proportion 2 of the Sixthbook of Elements, in which is shown, for the first time, the famous theoremattributed to Thales of Miletus according to whom a sheaf of parallel straightlines intersected by two transversals determines, on the two transversals,pairs of segments directly proportional.

6. Se fussero do linee equali perpendiculare sopra ad una linea et fusseroequidistante, et da un punto se partissaro do linee et andassaro a l’estremitàde la più propinqua et due altre a l’extremità de la più remota, dico che, sele stremità de socto sono soctoposte al puncto, che l’estremità de socto dela più remota se representarà al puncto più levata che la stremità de soctode la più propinqua et, se le stremità de sopra soprastanno al puncto, quellade la più remota s’apresentarà più basso (If there were two identical linesperpendicular over a line and they were parallel, and from a point originatetwo lines and these go to the extremity of the nearest one and two other tothe extremity of the farthest one, I mean that, if the lower extremities are po-sitioned beneath the point, that the extremity under the farthest one presentsitself at a point that is higher than the extremity below the nearest one and,if the extremities that are above are positioned over the point, the one of thefarthest will appear to be lower), which explicitly resumes proposition 10of Optics (De aspectuum diversitate), in which is shown how farther partsof planes that are placed under the eye appear to be higher than the nearerparts.

7. Se fusse la linea recta in più parti equale devisa, et da quelle divisioni separtissero più line et terminassaro ad un puncto, farano nel dicto punctodeseguali angoli: le linee più brievi farano magiore angolo che le piùlunghe (If a straight line were divided into several parts, and from thesedivisions originate more lines which terminate in a point, they will produceunequal angles in this point: the shorter lines will produce wider anglesthan the longer lines), which explicitly resumes proportion 24 of the Firstbook of Elements, in which instead is shown how two triangles that havetwo equal sides and the angle which is between them unequal, do also havethe ‘bases’ unequal.

8. Sopra a la recta linea data in più parti devisa se un’altra linea equidistante

26

a quella se mena et da le divisioni de la prima se tira linee che termininoad un puncto, devidaranno la equidistante in una proportione che è la lineadata (Above the given straight line divided into more parts, if we draw an-other line parallel to it and from the divisions of the first (line) we draw lineswhich terminate in a point, these lines will divide the parallel (line) in a pro-portion which is the given line), which explicitly resumes proportion 21 ofthe Sixth book of Elements, in which is shown that plane shapes, havingangles equal, are similar.

9. Se nella superficie quadrilatera la diagonale se tira, quella deviderà la su-perficie in do parti equali, et se equidistante dai lati altre linee si mena,deviderà la diagona con doi lati et sé in una proporzione (If in a quadri-lateral we draw the diagonal, this will divide the area into two equal parts,and if we draw other lines that are parallel to the sides, these will dividethe diagonal with two sides and itself in a proportion), which explicitly re-sumes proportion 34 of the First book of Elements, in which is shown theproperties of the diagonal of a parallelogram, and also proportion 24 of theSixth book of Elements, in which is shown how dividing a parallelogramby means of lines that are parallel to the respective sides, parallelogramssimilar to the first one will take shape, with which they have the diagonal incommon.

10. Se nella supeficie quadrata de equali lati et angoli equali la linea diagonalese mena, la quale se devida in più parti equali, et da quelle parti se menilinee equidistante ai quatro lati de la supeficie, deviderà quella in supeficiesimili (If in a square shape with equal sides and equal angles we draw thediagonal line, which we divide into several equal parts, and from these partswe draw lines that are parallel to the four sides of the area, this will divideit into similar shapes), which explicitly resumes proportion 21 of the Sixthbook of Elements, in which is shown how plane figures that have equalangles are similar.

11. Quando fusse uno quadrilatero che fusse in più parti deviso non equali,et da quelle parti se menassero linee equidistante ai lati, et dagl’angolise menasse la diagonale, deviderà quelle in proportione; et se da quelledevisioni se menassero linee per lo traverso che fussero equidistante, pro-durieno supeficie proportionali fra loro (If there were a quadrilateral thatwere divided into several not equal parts, and from these parts were drawnlines parallel to the sides, and from the angles were drawn the diagonal,this will divide them proportionally; and if from these divisions we drawcrossing lines that were parallel, these will produce shapes that are propor-tional among each other), which explicitly resumes proportion 22 of the

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Sixth book of Elements, in which is shown how, given four straight lines inproportion, shapes constructed on these are similar.

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[11] G. Monge. Géométrie Descriptive - Lecons données aux Écoles normales,l’An 3 de la République; Par Gaspard Monge de l’Istitut national. Baudouin,Imprimeur du Corp législatif et de l’institut, Paris, 1799. anastatica di Gabay.

[12] V. Sorge. La verità della visione: nota sulla ‘perspectiva’ di biagio pelacanida parma. DOCTOR SERAPHICUS, XLIV:51–64, 1997.

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