David A. Vila DominiScott Sutherland School of Architecture Robert Gordon University Aberdeen AB10 7QB UNITED KINGDOM
Despite the criticisms that Alberti made of Vitruvius, a large
amount of the detailed information that he provides is directly
transcribed from the older treatise. For example, although the
actual proportions of the columnar orders [4] as described in Close correspondences between the two treatises are frequent,
and it would seem correct in principle to assume that Alberti
took from Vitruvius what he considered to be of relevance in
some way to the kind of architecture that he was promoting. This
pattern of borrowing from earlier written works was to continue
through the proliferation of architectural treatises, including
Palladio's This paper proposes to look at one of those instances in which the three treatises run in parallel. The particular point I shall discuss is the recommendations regarding optical adjustment of the columnar diminution. In other words, the variation in diminution of column thickness according to the height of the column, and its implications for our understanding of the various practices with regard both to columnar proportion and visual sensibility in Antiquity and the Renaissance. The classical column traditionally comprises three elements: the base, the shaft and the capital. These take different forms according to the order to which they belong, Tuscan, Doric, Ionic, Corinthian, or Composite.[7] Although Alberti discusses the Tuscan temple in his treatise, he never mentions a column that would correspond to this order.[8] The base is absent in the Greek Doric, and its capital is almost seamlessly developed from the shaft, but most other columns conform to the base, shaft and capital schema.[9] The column shaft [10]
is an approximately cylindrical, vertical element which is wider
at the bottom than at the top. This narrowing of the column at
the top or, to be more precise, the difference or ratio between
the bottom and top diameters of the column shaft, is termed the
diminution.[11]
The profile of the shaft rarely, if ever, runs in a single straight
line, and in general, will have a kind of collar projecting just
below the capital, and another projection where it meets the
base -Greek Doric columns lacking this lower projection. Most
of the length of the shaft between the extreme projections has
a profile in the shape of a slight curve, which gives rise to
a swelling, or
Vitruvius begins his discussion of the diminution of the column shaft in the context of the arrangement of columns around a temple cella. Although in the text above it is not clear whether these observations about the diminution relate to all orders or only some of them, in Book IV iii on the Doric order he writes: "The column is to be diminished as directed for the Ionic order in the third book" [13]. Since for Vitruvius the shaft of the Corinthian column is the same as the Ionic one, it would seem that he saw these diminution ratios as applicable to the three orders. He relates the diminution to the adjustments made to the columnar orders in order to take account of optical effects. Vitruvius distinguishes between different types of optical effect. He has previously been talking about the columns on the corners of a temple facade: "The angle columns also must be made thicker by the fiftieth part of their diameter, because they are cut into by the air and appear more slender to the spectators. Therefore what the eye cheats us of, must be made up by calculation." Here it is not distance that is considered as a cause for diminished appearance; the cause, as anyone who has looked at a column against the bright Mediterranean sky will attest, is the brilliance of the background creating glare and 'eating into' the volume of the column.[14] The passage then goes on to stress that the variations in the contracturae -- the term Vitruvius employs for diminutions -- of columns are also due to an optical effect:
Another translation renders the above passage thus:
Vitruvius then goes on to talk about the rules governing the contracturae, and sets out the proportion of the top diameter relative to the bottom diameter of the shaft according to its height. A table expressing the variation in the contracturae according to column height, as described by Vitruvius, would look like this:
It can clearly be seen that in the left hand set of figures the shaft heights increase in multiples of ten, excepting the first figure, which is fifteen feet instead of the ten feet that might be expected. In the right hand column, the numbers in the numerator of the ratio are one unit less than the corresponding number in the denominator; both numbers in the numerator and the denominator increase by half a unit for every new column height in the left hand set of figures. Vitruvius gives no explanation as to why the shafts of columns should contract at all; Alberti, on the other hand, starts his discussion on the diminution precisely by giving such a reason: he found this reason in Nature.
The way Alberti sets out these rules varies slightly from that of Vitruvius. In terms of textual context, Alberti is discussing generalities and common aspects of the three columnar orders, Doric, Ionic, and Corinthian.[15] His intention is to express what these orders have in common, and to justify these common characteristics in terms of an understanding of natural laws: "As we mentioned in the last book, the Ionians, Dorians and Corinthians all favoured the same lineaments for their columns. Their columns have a further similarity, in that, as in Nature, the top of the trunk is always made more slender than the bottom" [Alberti 1988: VII.vi, p. 201]. The diminution, then, is a common trait throughout the columnar orders, for as he concludes, "in these respects, then, all orders agree", and it is in relation to nature that they share this trait. Here the trunk must be understood as a botanical metaphor, and not an anthropomorphic one.[16] Only after having expressed why it is natural for the column shaft to diminish does Alberti deal with the necessary adjustments in relation to the column height. The values are almost an exact transcription of the Vitruvian ones, although Alberti does not explicitly acknowledge his source here, and can be tabulated as follows:
First of all, Alberti introduces the 4/5 ratio which does
not appear in
The translators note that Palladio really meant Vitruvius III.iii., but do not alert us to the differences from that text, and indeed, the sheer inaccuracy of Palladio's figures. To begin with, Palladio only provides proportions for three ranges of column height, rather than the five that Vitruvius and Alberti provide. From Palladio's text it should be possible to extend the series to calculate the diminution for columns from 30 to 40, from 40 to 50 feet, and so on. In fact, though, the resulting ratios do not form a regular progression. Taking p/P to be the diminution ratio of columns up to fifteen feet, Palladio's series may be represented as: which, extended for columns up to 50 feet high, gives: This cannot be a workable system because the diminution ratios
alternately increase and decrease as the column height increases,
in clear breach of Palladio's observation that "the longer
the columns are the less they diminish". We find that what
Palladio gives is but a jumble of the ratios Vitruvius recommends,
and Alberti repeats, as can be seen from a comparison of If indeed Palladio believed he was giving the method according
to Vitruvius, then the inaccuracy of the transcription raises
the question as to how his figures come to be wrong. The error
could be ascribed to a copyist, but the extent of the differences
between the Vitruvian and Palladian versions seems too great
for a translator's or a scribe's work to be the sole cause of
the error. What does appear clear, though, is that Palladio cannot
have had a copy of
Alberti on the other hand, makes clear that, irrespective of its height, a column must diminish in accordance with natural laws; that is, columns are more slender at the top in the same way that Nature shapes tree trunks. For Alberti, therefore, the diminution is an intrinsic characteristic of the column, the reason for which is entirely unrelated to any consideration of the viewer. It is only after having made this distinction between the reason for column diminution, and the reason for the sake of which columns of different heights should have different diminution ratios, that Alberti proceeds to give the table of diminution ratios that he adapted from Vitruvius. In summary, therefore, it can be said that while for Vitruvius there existed a range of columns of different heights with different ratios of diminution, Alberti appears to be saying that there is a standard or original ratio of diminution, corresponding to a canonical or prototypical column, which is altered or adjusted to account for the observer's point of view. Alberti therefore makes a distinction between an aspect of the shape of the column (its taper) and the alterations and adjustments that have to be made to it on account of the changing viewing conditions resulting from the column's size [Vila Domini 1996]. This kind of distinction between what is permanently associated
with a body and what is only temporarily or accidentally so is
not a new one for Alberti to make. In his
For Alberti, the permanent properties of the surface in relation
to the art of painting are what he describes as the outline ( Vincenzo Scamozzi's additions to the
In order to explain Alberti's understanding of the implications of the diminution ratios given by Vitruvius, it is necessary to discuss in a little more detail some aspects of the fifteenth century invention of one-point perspective. In the perspective theory [20] set down in Alberti's De pictura (1435) following Brunelleschi's (1377-1446) discovery and experiments [See Manetti 1927: 9; Manetti 1970], the key is to identify the field of vision as a flat, rectangular projection plane which intercepts the rays of vision that proceed from the objects to the eye, or vice versa.[21] This theory is characterised by the single viewpoint of monocular vision, and employs the Euclidean principle of similar triangles. This method was probably arrived at in an artistic, building and surveying milieu, and, consequently, the projection plane takes on a very similar form to that of the surface of a painting or a window. Alberti described in De pictura the method by which to create a perspective projection, referred to as the construzione legittima or the veil. One of the concerns of perspective theoreticians such as Alberti and Piero della Francesca (1410/20-92), was to show that, when projected onto the perspective veil, proportional relationships in real or imaginary space were 'translated', so to speak, into corresponding proportional relationships on the two-dimensional plane. These were derived from the spatial ones by a ratio determined by a constant rule. In architecture, such a proof would also reinforce the idea that the eye was able to read 'spatial mathematics' and, therefore, would be able to appreciate the harmonic proportions Quattrocento architects wished to employ. In his article "Brunelleschi and Proportion in Perspective"
[Wittkower 1978: 124-136], Rudolf Wittkower clearly illuminates
the geometrical and mathematical principles that were employed
in order to explain this. Wittkower examines the development
of the theory that was aimed to prove that dimensional relationships
in space resulted in corresponding mathematical relationships
when projected, through the perspective cone, onto a flat plane,
such as the veil, a window, or a painting. The whole of this
theory is based on the proportionality of similar triangles found
in Euclid [ Some Renaissance perspective theoreticians endeavoured to
find the rate by which objects diminish in perspective. Alberti
did not give a rule regarding the ratios by which "a number
of objects in space diminish for the eye of an observer"
[Wittkower 1978: 128] . In his According to Wittkower, the particular case of this model that Leonardo [23] later examined must lead one to conclude that Brunelleschi and Alberti would have known [24] about the method by which to calculate the ratios of objects receding in space. Piero's series can be written as: n/(n+1).Because this series expresses the projected size of an object 'n' in terms of the projected size of object 'n-1', the actual values of the projected sizes [25] are given by the following series: nMathematically, this series tends to zero as the distance of the object from the eye approaches infinity. This means, of course, that the projection on the intersection becomes smaller and smaller the further the object is from the eye, though it decreases at a smaller and smaller rate also. Returning now to the diminution ratios recommended by Vitruvius and Alberti, it is possible to rationalise the expression of the ratios so as to write a table equivalent to the two above as follows:
It can now be seen that the ratios in the right hand column
belong to the generic series that in modern mathematical notation
can be expressed as
1/3, 2/4, 3/5, 4/6, 5/7, 6/8,7/9, 8/10, 9/11, 10/12, 11/13, 12/14, 13/15, 14/16, ... The value of the terms in this series The question then arises as to whether there might be a connection
between the Vitruvian series and those of Leonardo and Piero.
This question is especially tantalising since the modified Vitruvian
series responds to some sort of optical correction, the precise
nature of which is unclear, and the Leonardo and Piero series
set out the very basics of a means for representing optical effects
on a plane, one-point perspective. These apparent similarities
need careful analysis, for number series had been used for centuries
and for many different purposes. The fact that two series should
appear similar does not necessarily imply that the subjects to
which they are each connected are themselves related. But in
this case, it is possible to interpret the situation as a relationship
of inversion taking place. The Piero series, Antique painting had achieved considerable naturalistic realism
even in the representation of complex geometrical spaces such
as are found on surviving Roman wall paintings at Pompeii and
Herculaneum. Despite the clear evidence in some of these murals
of the use of construction points very similar to Alberti's One of the traits of the Renaissance perspective method of
projection is that the size of objects depends on the distance
between the plane of the intersection and the parallel plane
that the object exists in; in other words, an object will appear
smaller on the intersection when its perpendicular distance from
the projection plane increases. This means that an object may
actually be further away from the view-point than another of
the same real size, say because the second object is directly
in front of the view-point whilst the first is further to the
right, but will appear represented as of equal size on the intersection
if they happen to be on the same plane parallel to the projection
plane. One-point perspective involves no convergence of any lines
parallel with the picture or projection plane -- vertical, horizontal
or otherwise. It is only lines that are not contained by a plane
parallel with the projection plane that converge towards a vanishing
point, and therefore reflect the decrease in size due to distance.
This is the case with the representation of the column ( As Alberti points out in the introductory remarks to the diminution
series, "objects appear smaller, the farther they are from
the eye, [so the ancients?] sensibly [
Another question one may ask is whether the diminution series was devised with a particular order in mind. This may well have been the case, but if so, the numbers themselves offer little to uncover which particular order it may have been. In fact, because the diminution values are expressed as ratios, we find that it is impossible to tell whether they relate to slender or sturdy columns, and therefore making it impossible on this basis alone to determine whether they may relate to Doric or Ionic (and Corinthian) type shafts. Curiously, because the dimension of the upper diameter of the shaft is given as a proportion of the lower diameter, if a thicker column -- i.e., one with a larger width to height ratio- were designed using the Vitruvian diminution ratios, it would have a shaft whose outline is more prominently inclined than that of a slender column. This peculiarity responds pretty closely to the reality of the outlines of Ancient Greek Doric examples when compared with Ionic, or Hellenistic and Roman Corinthian, where these last ones are composed of straighter shafts. In other words, the Vitruvian diminution ratios would initially seem equally well suited to describe a shaft belonging to any order. Better access to the reality of Renaissance buildings ought to make obtaining detailed survey information easier, although the measurement of the upper diameter necessary to establish the diminution is not one of the most usual ones taken. As well as the buildings and the texts, the Renaissance has left a wealth of drawn material from which in principle it ought to be possible to extract some information regarding the diminution of columns. The reality is that the type of engraving furnished in Palladio's treatise, for example, is difficult to interpret with much accuracy, though tentative studies I have made tend to suggest columns less diminished than the recommended by the Vitruvian series. Alberti produced no drawings in his ten books, and one might have to rely on the text for information as to his view on the diminution because most of his built works were executed by others who had an uncertain degree of independence from Alberti himself. In fact, if Alberti intends architects to adjust the diminution to the height of the column, and does not recommend the ratios he copies from Vitruvius, then there is no way of knowing from his treatise what he intended his contemporaries and followers to do. When Alberti, a little later on in the Ten Books, comes to describe in detail the setting out of a column shaft, he does not follow the series he transcribed from Vitruvius, but decreases the amount of the diminution considerably by using different ratios. In VI.xiii, he says: "according to the size of column (a subject to be debated in the appropriate place) the length of the diameter of the top circle is derived from that of the lowest circle..."[28]. The appropriate place is either VII.vi, where he gives the aforementioned series, or, a little below, where he describes a thirty foot high column: The maximum overall diameter is divided into nine equal parts, of which the diameter of the projection at the top takes up eight, giving it the ratio nine to eight, called the sesquioctave.[29] Now, this 8/9 ratio would correspond, by an extension of the Vitruvian and Albertian series, not to a thirty foot column, but to a column of a height up to seventy feet -in the series they both give, columns of up to thirty feet should be constructed with the ratio 6/7. Applied as it is by Alberti, to a thirty foot column, it gives a smaller diminution than if he had followed the diminution series. This smaller diminution ratio is very possibly consistent with the Roman examples he claims to have measured. Alberti was probably not an exception in reducing the amount by which the shafts were diminished; the Spanish eighteenth-century translator of Vitruvius, Ortiz y Sanz, was of the opinion that greatly diminished columns, such as those recommended by Vitruvius, were ungainly to look at.[30] But what is puzzling is that in his treatise Alberti should have set down the diminution series and then this example which ignores it, almost side by side, without telling us the reason for disregarding the former and adopting the latter. Columns were certainly diminished in Palladio's time, and
indeed in Palladio's own work, but the fact that he gave such
a jumbled version of the diminution ratios in itself may shed
some light on his personal view on how the diminution was worked
out in practice. If Palladio used the Vitruvian diminution method
in the designs of his own buildings, it would seem reasonable
to expect him to have a better recollection of the figures involved
when he came to set them down in the
As we have seen, the method originally employed to calculate the ratios given by Vitruvius in the diminution series cannot have been the one used by Alberti and his contemporaries to construct perspective projections. Indeed, this type of perspective could not register any change in size due to increased distances from the viewing point, if these increases were due solely to height. Therefore, it is necessary to conclude that a different method was used in antiquity in order to arrive at these ratios. This method is likely to have been derived from the science of optics, on which Euclid was an authority, or from some application of it, perhaps following Richard Tobin's suggestions [32] of the possible development of a system of perspective projection based on a spherical projection plane. Any such method would have made it possible to detect increases in distance due to height, and the consequent apparent reductions in the size of objects. A further, perhaps simpler, method would be one reliant exclusively on the ratios of distances from the viewing point to the top and bottom of the column, for, with a previously established viewing position, it would be a relatively easy task to calculate arithmetically and graphically the proportionate reduction in size of the top diameter, and adjust it so as to achieve a measurable perceptual ratio with the bottom diameter. The actual method employed in antiquity would be dependent on an elusive 'correct' or desirable way of, or attitude towards viewing a facade composed of columns. The existence of a precise method for adjusting the appearance of the columns' top diameter must have responded to a particular appearance that was sought, or to a particular effect that was to be avoided. One would be inclined to think from the remark "what the eye cheats us of, must be made up by calculation", that this desired or undesired appearance was itself subject of being measured. In order to understand the ratios it would appear necessary, therefore, to develop a model of viewing a building which may have applied in antiquity. I would suggest that a starting point for such future investigation in regard to the diminution of columns should be to determine what optical effects the Vitruvian diminution ratios actually have on the perceived ratios of top to bottom diameter when employed to govern the proportion of columns. Furthermore, another question that arises regarding the diminution, is what might have been Alberti's and Palladio's intentions in providing this information, and, particularly, whether they intended architects to use these tables or not. Alberti subscribes to the view that architecture, having originated in Asia and acquired beauty in Greece, was finally perfected by the Romans.[33] As it is quite clear to him, from his own surveys, that the kind of diminution recommended by Vitruvius was not in use amongst the Romans, he can only have regarded this as an example of Vitruvius's Grecophilia. But unfortunately, in this passage on the diminution Alberti does not give what rule, if any, the Romans did employ. The diminution series is not the only example of optical adjustments given by Vitruvius and Alberti. Amongst other optical adjustments, their recommendations regarding the height, or thickness of the architrave [34] in relation to column height constitute a very similar case to that of the column diminution. The essence of the matter is the same, that is, the higher the column, the thicker the architrave is required to be in relation to the column diameter (and height), in order to compensate for the reduction of perceived size due to the increased distance (height) from the eye at which the architrave is positioned. I will not enter into a detailed examination of this, but it can readily be seen from the recommended ratios that the situation is much the same as with the column diminution, and most of the questions that apply to one are, therefore, also relevant to the other. To conclude, then, we have seen that the original source of the diminution ratios might have been known to Vitruvius. But whether he, Alberti or Palladio understood the method by which these ratios were derived, or their precise effect on the visual appearance of columns, is something that must remain questionable. At the same time, I have pointed to the need for more research to be carried out in relation to ascertaining the optical effects of using the diminution ratios. Any such research should probably begin by examining the mathematical nature and characteristics of the figures involved, and the optical effects when viewing columns designed using these ratios. APPENDIX
I : VITRUVIUS ON THE DIMINUTION OF THE COLUMN SHAFT
APPENDIX
II : ALBERTI ON THE DIMINUTION OF THE COLUMN SHAFT
[1] Some of the material in this paper has previously appeared in the author's "The Diminution of the Column in Alberti and Vitruvius; Concerning Optical Corrections" [Vila Domini 1996] and "Column Marginalia" [Vila Domini 2000]. return
to text[2] "For I grieved that so many works of such brilliant writers had been destroyed by the hostility of time and of man, and that almost the sole survivor form this vast shipwreck is Vitruvius, and author of unquestionable experience, though one whose writings have been so corrupted by time that there are many ommissions and many shortcomings. What he handed down was in any case not refined, and his speech such that the Latins might think that he wanted to appear Greek, while the Greeks would think that he babbled Latin. However, his very text is evidence that he wrote neither Latin nor Greek, so that as far as we are concerned he might just as well not have written at all, rather than write something that we cannot understand" [Alberti 1988: VI, i, p. 154]. return to text[3] I have told you that I desire to make my language
Latin, and as clear as possible, so as to be easily understood.
Words must therefore be invented, when those in current use are
inadequate; it will be best to draw them from familiar things.
We Tuscans call a fillet the narrow band with which maidens bind
and dress their hair; and so, if we may, let us call 'fillet'
the platband that encircles the ends of the column like a hoop.
But the ring positioned at the top nest to the fillet, which
binds the top of the shaft like a twisted cord, let us call 'collar'
[Alberti 1988: VI, xiii, p. 186]. [4] The words used by Alberti and Vitruvius to refer
to these is not 'orders', but others such as [5] Referring to his description of the column shaft,
Alberti writes that it "is not a discovery by the ancients,
handed down in some writing, but what we have noted ourselves,
by careful and studious observation of the works of the best
architects" [Alberti 1988: VI, xiii, p. 188]. [6] See Alberti [7] For Alberti, the only difference between the Corinthian
and the Composite (which he refers to as Italian) columnar styles
is the make-up of the capital; Alberti rejects other types of
column style in [8] Those who claim that Alberti sets out the five columnar
styles are mistaken. For example, Peter Murray writes: "Alberti
gives the first consistent theory of the use of the five orders
since classical times" [Murray 1986: 53]. In De re aedificatoria
VII vi, Alberti talks about three main types of capital, the
Doric, Ionic, and Corinthian, and he adds the Italian capital
as a variant worthy of inclusion because of its beauty. Although
he claims that the Doric was used in ancient Etruria, perhaps
before the Greeks used it, he does not identify a separate Tuscan
capital. Rather, for Alberti, the Tuscan and the Doric seem to
be one and the same thing. When describing a monumental column
in De re aedificatoria VIII iii, Alberti comes close to Vitruvius's
Tuscan column, but calls it Doric. [9] For Alberti, the Doric column has a base, and so
all columns consist of the three elements: base, shaft and capital. [10] Both Alberti and Vitruvius refer to the column shaft
and to the whole column including base and capital with one word,
columna, which makes it difficult on occasions to be sure whether
the measurements or proportions they describe relate to the whole
element or only to the shaft. [11] "DIMINUTION- The amount of tapering or reduction
in the diameter of a column shaft from bottom to top, generally
two-ninths to one-fifth of the lower diameter in Doric and one-sixth
to one-seventh of the lower diameter in Ionic and Corinthian
columns" [Dinsmoor 1950: p. 390]. It should be noted here
that Dinsmoor expresses the diminution as the [12] Vitruvius appended an illustration exemplifying
the method by which to draw the entasis, but this had been lost
by the time Alberti read him. ( [13] [15] Cf. notes 8 and 9 above. [16]
[Alberti 1972: 136-137]. [17] On the Renaissance preference for whole numbers
see "The Changing Concept of Proportion" [Wittkower
1978: 109-123, esp. 111-116]. [18] [19] [20] For a good introduction to perspective see [Kemp
1990], especially Chapter I. [21] On rays of light proceeding from the object or the
eye, see note 14 above. [23] Leonardo da Vinci simplified the problem of the
perspectival diminution of objects by considering a particular
case of Piero's model, in which he made the first object coincident
with the projection plane, and at one unit distance from the
eye. This model allows us to express the size of the projection
of object [24] "One must take it for granted that artists
had known about the simple proportion [...] ever since similar
triangles had been used in the context of perspective, and from
this one is bound to infer that Leonardo's progression can never
have been a secret" [Wittkower 1978: 132]. [25] See note
22 above. [26] The series is equivalent to the one that can be
drawn from the table given by Vitruvius, except that both the
numerator and denominator are doubled. The series arising from
Vitruvius's table is ( [27] The angle of vision had to be kept below a certain
limit if the objects and spaces at the edges of the painting
-or at points on the painting distant from a central vanishing
point- were not to appear uncomfortably distorted (for a concise
discussion of this problem in relation to a row of columns, see
[Gombrich 1972: 215-216]). But, in turn, small angles of vision
would often mean longer viewing distances, which would have been
laborious to construct in practice. Artists were aware of these
undesirable effects and endeavoured to compensate unsightly distortions
in a number of ways, such as limiting the angle of vision. [28] In [Alberti 1988: VI.xiii, p. 188], the translation
of [29] [Alberti 1988: Book VII.vi, pp. 201-202]. [30] [31] Hermogenes was one of the sources cited by Vitruvius,
[32] [Tobin 1990]. In this article, Tobin analyses possible
misreadings of Theorem 8 in Euclid's Optics, and reconstructs
a hypothetical spherical projection method that the Greeks might
have been able to develop following Euclidean geometry and optics.
[33] "Building, so far as we can tell from ancient
monuments, enjoyed her first gush of youth, as it were, in Asia,
flowered in Greece, and later reached her glorious maturity in
Italy" [Alberti 1988: VI.iii, p. 157]. [34] "They therefore laid down the following rules: for columns up to twenty feet, the height of the beam should be one thirteenth that of the column; for those up to twenty-five feet, a twelfth; then, for columns up of to thirty feet, the beam should be one eleventh of that height; this same progression was then used to calculate the remainder" [Alberti 1988: VII ix]. Cf. [Vitruvius 1955: III.v]: "The proportion of the architraves should be as follows: if the columns are from twelve to fifteen feet, the height of the architrave should be half the thickness of the column at the bottom; from fifteen to twenty feet let the height of the column be divided into thirteen parts, and the height of the architrave be one part; from twenty to twenty-five feet, let the height be divided into twelve parts and a half, and let the architrave be one part of that in height; also from twenty-five to thirty let it be divided into twelve parts, and let the height be made of one part. Thus the heights of the architraves are to be determined in accordance with the height of the columns."
return
to text
L'Architettura
(De re aedificatoria). ed. & transl. Giovanni Orlandi,
ed. and trans. Introduction by Paolo Portoghesi. Milan: Il Polifilo.Alberti, Leon Battista. 1972. Alberti, Leon Battista. 1988. Dinsmoor, William Bell. 1950. Gombrich, Ernst.1972. Grayson, Cecil. 1960. The Composition of L.
B. Alberti's 'Decem Libri De re aedificatoria'. Kemp, Martin. 1990. Krautheimer, Richard. 1963. Alberti and Vitruvius.
Pp.42-52 in Manetti, Antonio. 1927. Manetti, Antonio di Tucci. 1970. Murray, Peter. 1986. Ortiz y Sanz, Joseph, ed. and trans. 1987. Andrea Palladio. 1997. Pollit, J.J. 1974. Tavernor, Robert. 1991. Tobin, Richard. 1990. Ancient Perspective
and Euclid's Vila Domini, David A. 1996. The Diminution
of the Column in Alberti and Vitruvius; Concerning Optical Corrections.
Vila Domini, David A. 2000. "Column Marginalia",
Vitruvius. 1955. Vitruvius. 1960. Wittkower, Rudolf. 1978. The Changing Concept
of Proportion. Pp. 108-123 in White, John. 1956.
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