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In architectural treatises compiled from the Renaissance up through the last century, the study of the Ionic volute has been set forth and debated innumerable times, and many diverse solutions have been elaborated and propagated, so much so that the theme of the correct method for laying out the volute retained a significance that was far from marginal up through the nineteenth century.[1] The debate that surrounded the form of this element began between the 1400s and the 1500s, and arose from the difficulty of interpreting a brief passage in the Roman architectural treatise of Vitruvius.[2] This difficulty posed problems of a practical as a philological nature: it was not only a matter of establishing the correct interpretation of an antique and authoritative text, but-much more importantly-the object was to develop (based on the authority of Vitruvius) a method for laying out the volute that could then be put into practice in the construction methods of the 1500s, thus permitting the general use of the Ionic order. Our interest revolves around the characteristics of three methods for laying out the volute-each of which would subsequently enjoy great notoriety-developed by three authors during a relatively brief period at the beginning of the 1500s. These methods have been studied with regards to their geometric and mathematical characteristics, beginning with the original texts in which they are described. First they were redrawn, in order to study how straightedge and compass were utilized, then the mathematical properties that underlie them were studied, and finally their eventual classification according to types of spiral were verified.
The study of ruins was concentrated mainly on the ruins of
Rome itself, already well known to the architects and artists
of the 1400s. Initially, their knowledge was restricted to the
visual observation and to studies The Ionic order, with its characteristic capital, is one of
the Les Edifices Antiques de Rome, Paris 1682.The discussion in Vitruvius's treatise that regards the Ionic order and its various parts (including the spiral volutes) is concentrated on a brief passage of text in which Vitruvius describes the Ionic volute as a curve in the form of a spiral, which, starting at a point immediately under the abacus (the upper, flat, part of the capital upon which rests the architrave) winds in a series of turns until it joins a circular element known as the "eye". The center of the eye is situated on a vertical line, parallel to the cathetus, the line that descends vertically to the extreme point of the abacus [Vitruvius 1997: III, 256]. The Vitruvian text furnishes some basic rules that are very clear, but it is more difficult to interpret the real laying out of the curve of the volute, above all beyond the first turn. Vitruvius only furnishes proportional ratios, and no concrete dimensions. The base rules indicated in his text, which we have also indicated in Fig. 1, are: - Total height of the volute: 8 units;
- Diameter of the eye: 1 unit;
- Above the center of the eye: 4 1/2 units;
- Below the center of the eye: 3 1/2 units [5]
Vitruvius then simply indicates that the volute is laid out by means of quarter-circles, using a compass [6], starting with the outermost point, up to arriving at the conjunction with the eye, decreasing by one-half unit the distance from the center of the eye at each successive quarter-circle set out. The text does not indicate precisely how many quarter-circles are necessary to complete the volute, nor does it indicate the position of its center, nor does it indicate how many turns must be completed in all. If the diminution of the radius of one-half unit every quarter-circle remains constant, it is evident that eight quarter-circles are sufficient to reach the eye,[7] that is, two complete turns, a detail that is missing from the text. Vitruvius makes reference in his text to a figure that has not come down to us. The lack of the drawing (which probably was only an elementary geometrical scheme [8]) introduces an element of the unknown (only apparently marginal) in the description of the volute beyond the first turn, furnishing for Renaissance theorists both a starting point and a justification for the development of their own methods. To this unknown element were subsequently added, as the surveys of the Roman ruins proceeded, the contradictions between the surveys and Vitruvius's treatise. In the case of the Ionic capital, measuring and surveys make evident the fact that the capitals possess volutes in which the distance between the turns diminishes progressively towards the center and in which the turns are usually three, partly contradicting, or at least rendering more obscure, Vitruvius's descriptions. Further, between capital and capital there are significant differences in measurements and forms.[9] The methods developed during the Renaissance represent a reaction of the Renaissance culture to this situation of uncertainty and contradiction. Each author goes in search of his own model of the Ionic capital, derived from his own personal critical interpretation of the literary and archaeological sources, using drawing as an instrument for formal analysis.[10] The aim of the research into antiquity is progressively transformed from mere imitation to the search for and invention of design rules for application in actual projects [Günther 1994, 266].
- the method of semi-circular arcs of Sebastiano Serlio (1537);
- the method of quarter-circle arcs of Giuseppe Salviati (1552);
- the method of eighth-circle arcs of Giullaume Philandrier (1544).
In order to compare these constructions we have chosen to redraw them according to the instructions in the original texts, utilizing a modern program for assisted architectural drawing. This procedure presupposes on the one hand the accurate comprehension of the method, annulling errors of draftsmanship and approximation; on the other hand it allows us to extract directly from the drawing a set of numeric data that is highly precise, which we will use to set forth a first series of observations, on the basis of which we will proceed to the mathematical analysis of the construction.
In order to describe the numerical order of the centers, which
is the most important information contained in Serlio's method,
he refers explicitly to a figure [13] in which, however, the numeration is
indicated in a somewhat dubious, if not erroneous, manner (rather
than 1-3-5- The volute drawn by Serlio has the characteristic of making
three complete turns before closing on the eye. The method of
six points is very ingenious in its simplicity, and probably
owes something to Leon Battista Alberti, who in his
Salviati [1522] writes of having rediscovered the method after undertaking a precise geometric study and starting from a simple enough observation. If, as Vitruvius indicated, after a quarter turn the distance of the center diminishes by one-half unit, then if only an eighth of a turn is made, the distance of the center should diminish by only one-quarter unit. Fixing these three points (the initial, that of the eighth turn, and that of the quarter turn), he draws by means of them the segment of a circle, using, as he clearly indicates in his text, "the fifth proposition of the fourth [book] of Euclid", that is, the theorem that establishes how through any three distinct points there can pass only one circumference. The Euclidean text in effect deals with the problem of circumscribing
a given triangle with a circle, and is therefore equivalent to
the determination of a center of a circumference of which three
points are known. Salviati brings to light in any case a problem
(which will be partially evaded by the successive propagators
of his method): two successive centers must turn out to be aligned
with the point of contact between their respective circular arcs.[18] If this
doesn't occur, the curve appears as though "broken"
and the volute is not a smooth curve. Here again Salviati makes
explicit reference to the Since Vitruvius does not indicate the diminution of the volute
beyond the first turn, and since, maintaining the diminution
of one-half unit each quarter-turn, the volute has only two windings
with regular intervals between the turns, Salviati proposes for
the second turn a diminution of one-third unit every quarter
turn, and for the third turn a diminution of one-sixth unit for
each quarter turn, proceeding then to the determination of all
the centers, formulating thus his rules of construction [Salviati
2000: 106].
To lay out Philandrier's volute, it is necessary to execute a preparatory drawing of a right triangle ABC with base 3 1/2 units and height 4 1/2 units. The eye is centered in the right angle C. Vertex A is connected with the point of intersection between cathetus BC and the eye. An arc is drawn by centering the compass in A and opening it equal to AC. The segment of the arc between the hypotenuse and the segment previously drawn is divided into 24 equal parts. The points thus obtained are projected with the center at A on cathetus BC, distinguishing 24 points. These points are traced onto the main drawing, along a series
of lines set out starting with the center of the eye at 45°
intervals, and determine the 24 points of the volute. The points
are joined with circular arcs, for which, however, Philandrier
indicates neither the precise locations of the centers nor a
way of determining these centers. It is obvious that they can
be found using the Euclidean method already discussed by Salviati.
The result is a very rounded volute, which closes on the eye
after three turns.
The difference in process between the Serlian volute and that of Salviati is actually minimum and the two volutes coincide in many points when superimposed. If we observe them separately we have the impression that the Serlian volute leans rather to the right, but this is actually an optical effect. The volute of Philandrier differs from the other two in a fairly clear way, especially in the second, larger, turn, and reaches the eye only with an accelerated final turn. We have composed a table of values of fullness of angle a (in radians) and the length of the radii
r(a) in function of the angle for
a series of common points. By radius we mean the distance of
the points of the volute from center O of the system of polar
coordinates. The values of r(a ) were
measured directly from the computer drawing, with an approximation
to three decimal places.[22]
It should be recalled that we are dealing with a graphic method;
an algebraic approach is not interesting
Table 1. Three constructions compared Our next step was to transfer the data from the table to a
Cartesian graphic, with the values of on the abscissa ( The rate of increase of the radii of the volute of Philandrier-Dürer differs from the other two in the first two turns. It has an initial growth that is much more regular, even if not linear, and therefore more accelerated as increases. The way in which the radii increase can help us to understand if curves that are interesting from a mathematical point of view are hidden within these three systems, and if it is worth verifying whether or not these volutes correspond, approximately or rigorously, to known equations for spirals. In the diagram in Graph 2 we have represented the increase (the difference between consecutive radii) of the radii of the volutes. Graph 1 shows clearly how the increases in the radii of the
volutes of Serlio and Salviati differ for each successive turn.
In contrast, the increase of the radii is almost constant on
the interior of each turn. The equation of a spiral curve in
which the increase of the distance from center O is constant
is that of an Archimedean spiral, r(a)
= Graph 1 and Graph 2 show that the search for a correspondence
between single equations of known spirals and the curves drawn
by Serlio and Salviati is relatively meaningless. Graph 1 indicates
that every hypothetical curve capable of approximating with a
greater or lesser degree of precision the points of Serlio and
Salviati must have this characteristic: its growth (by our convention
the volute grows from the eye to the exterior) should be constant
and convex in the interval between p/2
and 13p/2 (the distance of the turns
increases as it goes further away from the eye). The number of
possible equations that satisfy this condition is infinite. For
example, it is possible to determine parameters The situation with Philandrier's volute is different. Graph 2 shows a progressive increase in the radii, but not in steps. The construction does not appear to be composed of distinct parts, and it is interesting to try to understand if it can be more or less closely approximated by some type of spiral. In the original treatise by Dürer, the construction appeared after the method for constructing the Archimedean spiral, as a variant of it, for when the object is to construct spirals that become ever more compressed towards the center. Its construction, adopted by Philandrier for the Ionic volute, is in fact, in contrast to those of Serlio and Salviati, constructed by points which are connected by circular arcs, which, as Dürer noted, can even be drawn freehand. Dürer's procedure is much more generalized: the treatise presented a spiral constructed from twenty-four points which could be divided either by twelve radii distributed in two turns, or by eight radii distributed in three turns-thereby keeping open the option of the number of turns in which the volute must close. Further, the number of fixed points of the layout can be increased at will, increasing the number of the subdivisions of the arch in the triangle (but requiring then the layout of a greater number of circular arcs that are increasingly smaller to connect the points). Dürer himself further indicated that it would be possible to construct a spiral that narrowed or widened towards the center by varying the inclination of the cathetus of the triangle, transforming the angle from right to obtuse. Dürer, who was not looking for a method of constructing an Ionic volute, is principally interested in the construction of spirals. Thus, the construction propagated by Philandrier is a particular case, applied to the Ionic volute, of a whole series of possible spirals. Completing the preparatory construction in Fig. 4 with some supplementary instructions allows us to make some observations. At this point it is necessary to calculate the values of the radii beginning with angles and , obtaining: and so forth. Generalizing for a whole
The generating function of this construction is therefore a trigonometric function. It is important to note that even if formula (1) appears to be a function of the number of radii and not of the angles, this is not the case, since each radius corresponds to a precise and determined angle . In fact, it is possible to generalize formula (1) with a real
number between p/2 and 13p/2,
transforming the series n-1) with ,
which indicates the rotation of p/4
of the radii, is substituted by (a
- p/2) with .
The equation in polar coordinates of the volute of Philandrier/Dürer
will thus be:
The turns that the spiral makes before reaching the eye are
defined as denominator of the argument with the value 6p = 3*2p = 3 turns. An increase or decrease of
this value makes the spiral make more or fewer turns, that is
In any case, if two methods were shown to be effectively only
geometric constructions, which could certainly be approximated
by means of functions but which were not themselves approximations
of particular curves (neither by the intention of their creators
nor in their characteristics), the third method showed characteristics
that were interesting from a mathematical point of view as well. Instead, Salviati and Serlio were not searching for a method for approximating spirals, but only for a way of laying out a drawing that would mediate between what academic studies of Vitruvius and direct studies of the ancient monuments brought to light. Geometrically speaking, Salviati and Serlio are interesting above all for the determination of the centers of the circular arcs (a problem derived from Vitruvius) that constitute the curve. The methods don't refer to any particular known spiral, but are only imitations of the idea of spirals, conceived to suit the Ionic volute, in contrast to that of Dürer, which was much more general. From a mathematical point of view we can affirm that they are constituted of three truncated Archimedean spirals, and that infinite continuous curves exist that are capable of approximating them in three complete turns. Their geometric nature does not permit generalizations, but it makes them easy to comprehend and immediately applicable. In particular, the construction of Salviati enjoyed an enormous success, and was republished in innumerable architectural treatises from the 1500s to the 1800s.[24]
[2]
The [3]
Between the middle and the end of the Quattrocento Rome was the
center of an intense program of study of antiquity promoted by
Pope Leo X. The protagonists of this movement were Bramante,
Simone Pollaiolo (called Cronaca) and Guiliano da Sangallo. The
other vestiges of antiquity, strewn about the territory of the
ancient empire (especially those outside Italy, but the ruins
of Magna Grecia in Campania and Sicily as well) were unknown
or ignored for a long time. See [Milton and Lampugnani 1994;
Günther 1994; Agosti and Farinella 1997]. [4]
Temples in the Ionic style were built in Greece and Asia Minor
from 550 B.C. onwards [Koch 1998, 10 ff]. The Romans used the
Ionic order in varying kinds of public buildings-temples (see,
for example, the Temple of Fortuna Virilis in Rome), baths, amphitheatres-as
well as private buildings such as palazzi. [5]
[6]
[7]
4.5 - 8*0.5 = 0.5, which is the radius of the eye. [8]
According to Mario Carpo, the other drawings cited in the Vitruvian
text, ten in all, are all elementary geometric schemes rather
than actual architectural elements [Carpo 1998, 22]. [9]
See [Ashby 1904], which discusses the so-called Coner Codex,
a collection of sketches from Roman antiquity with very accurate
dimensions, published in Rome presumably around 1515. A series
of plates also presents various Ionic capitals, the volutes of
which vary considerably from one to the other, but are usually
of three revolutions. [10] See [Recht 2001] on the role of drawing as a means
of bringing to life an ancient text and a centuries-old architectonic
culture. See also [Carpo 1998] on the role played by print publication
in the affirmation of drawing as a means of communicating ideas.
[11] The [12] Actually, this book, in spite of its denomination
"Book IV", was the first published in the series that
makes up Serlio's treatise. [13] [14] See for example the Venetian editions of 1566 and
1588, which give the erroneous numeration. Instead, the German
editions of Basel of 1609 gives the correct numeration. [15] However, the natural impulse is to trust the drawing,
postulating some kind of error. It is possible, in fact, to construct
an Ionic volute even from Serlio's "erroneous" construction;
in order that the approximation of the spiral closes on the eye,
it is necessary however to vary some of the dimensions given
in both Serlio's and Vitruvius's texts. If we indicate, for example,
the diameter of the eye as d _{0}) - 5/6 d_{0}
- 4/6 d_{0} - 3/6 d_{0} - 2/6 d_{0} -
1/6 d_{0} = 4 - 14/6 d_{0}Since, according to the construction, the
last radius must be of a length 4/6 d _{0} -1)/2] + 1/6
d_{0}) - 5/6 d_{0} - 4/6 d_{0} - 3/6
d_{0} - 2/6 d_{0} - 1/6 d_{0} = 4 + 1/2
- 17/6 d_{0}Setting again the minor radius equal to 4/6
d [16] [Alberti 1483: VII, viii]. The larger circle measures
8 units, the eye 1 unit. The center of the eye is lower by 1
unit with respect to the center of the larger circle. There are
two points in which the compass is placed alternately: the upper
and lower points of the eye, beginning with the upper. Since
the radius of the half-circumference decreases by 1 unit every
time, with the first radius being 4, after four operations the
arcs end on the eye and the volute is complete. The volute has
only two turns and the distance between them remains constant.
In this the solution adapts itself well to the hypothetical volute
of Vitruvius, leaving aside a critical comparison with the ruins
of antiquity. It is interesting, however, to note the simplicity
of the Albertian method, in which can be recognized in part the
inspiration for Serlio. [17] [Kruft 1998, 35]. Alberti wrote in Latin for a cultured,
humanistic readership. His text was initially published without
illustrations. The [18] [19] It should be noted that the method developed by
Salviati is approximate as well. In fact, if we try to follow
his directions and try all the centers of the twelve quarter-circles
according to his rules (using a modern CAD program), we discover
that they are not only located on the diagonals of a square,
but that they are slightly rotated. We have, however, decided
to base our analysis solely on the directions given in the original
text. [20] This work enjoyed great fortune and numerous successive
editions were published; see [Lemerle 2000], a facsimile publication
of the edition published in Lyon in 1552, which also includes
a rich critical apparatus. [21] Dürer gives no indication as to how he developed
the methods he presents. An interesting comment on the text can
be consulted on the Internet, at the URL http://www.mathe.tu-freiberg.de/~hebisch/cafe/duerer/spiralen.html.
[22] Naturally, an algebraic calculation is possible,
even if in the case of Salviati it is rather lengthy, due to
the recursive application of the Pythagorean theorem with irrational
values. For Serlio's construction this is very simple: the length
of the radius is derived directly from the rules for setting
out. [23] For Salviati, we have a= 80/10000, b= 364/10000
and c= 4291/10000; for Serlio a= 80/10000, b= 354/10000 and c=
4404/10000. [24] Serlio's construction was abandoned after some time,
while the others were taken up again and again in successive
publications. The generation of a complete list of the use of
the methods of Salviati and Dürer/Philandrier on the part
of architectural theorists who have studied this is beyond the
scope of the present paper. Perhaps it is most interesting to
point out that in his
Michelangelo.
Studi sull'antichità dal Codice Coner. Turin: UTET.Alberti, Leon Battista, 1483. Ashby, T. 1904. Sixteenth-century drawings
of Roman buildings attributed to Andreas Coner. In Carpo, M. 1998. Chambers, W. 1825. D'Avilier, Augustin-Charles. 1760.
Desgodetz, A. 1682. Dürer, A. 1525. Forsmann, E. 1988. Goldmann, Nikolaus. 1696. Günther, H. 1994. La rinascita dell'antichità.
Pp. 259 ff. in Kruft, H.W. 1998. Johst, Chistopher. 1994. Lo studio dell'antico.
Pp. 425 ff. in Koch, W. 1998. Lemerle, F. 2000. Losito M. 1997. La ricostruzione della voluta
del capitello ionico vitruviano nel Rinascimento italiano. Pp.
1409 ff. in Vitruvius, Milton, H. and V. M. Lampugnani, eds. 1994.
Palladio, A. 1570. Philandrier, G. 1544. Recht, R. 2001. I Salviati, G. 1552. ______. 1814. Regola di far perfettamente
col compasso la voluta jonica et del capitello ionico et d'ogni
altra sorte. In G. Selva, ______. 2000. Scamozzi, V. 1615. Serlio, S. 1537. Schwarz, G. 1837. Die Schneckenlinie des ionischen
Säulenkapitäls. Vignola, J. Barozzi da. 1562. Vitruvius. 1997.
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