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Classical Greek and Roman Architecture:Mathematical Theories and Concepts
Sylvie Duvernoy
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2The Figurate Representation of Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
The Visual Comparison of Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8The Theory of Proportion and Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Musical Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12The Duplication of the Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Art and Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Abstract
In classical antiquity only round numbers — natural integers — were known,and mathematics was very different to the way it is today. But whereas themathematics of this ancient era was in one sense more basic, it made use ofmany theoretical concepts and approaches that are no longer familiar to modernscientists. This chapter introduces three mathematical concepts or approachesthat provided a foundation for classical Greek and Roman architecture. The firstof these, which was equally significant for geometry and arithmetic, is concernedwith the figurate representation of quantities. The second is associated with thevisual comparison of magnitudes, and the last is the theory of mean proportions.
S. Duvernoy (�)Politecnico di Milano, Milan, Italye-mail: [email protected]
© Springer International Publishing AG, part of Springer Nature 2018B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_61-1
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2 S. Duvernoy
KeywordsPythagoras · Euclid · Plato · Mean proportional · Musical proportions ·Commensurability · Symmetry
Introduction
The Classical era in Greece and Rome is conventionally regarded as spanning almost1000 years, from around 800 BC through to 200 AD. During this period the Greco-Roman world laid the foundations for contemporary thought in science, philosophyliterature, art, politics, architecture, and mathematics. The latter two are the focusof the present chapter, and in particular the mathematical theories, concepts, andpractices that became central to the architecture of the era, and continue to havean enduring, if more subtle influence to the present day. In order to understand therelationship between architecture and mathematics in classical antiquity, it is firstnecessary to understand the essence of Greek mathematics and the methodologiesof scientific research that made developments in mathematical and architecturalknowledge possible.
Geometry and arithmetic, the two main branches of ancient theoretical mathe-matics — and also two of the four liberal arts — relied on sensorial perception,both in the process of undertaking research and in the process of divulgation.Sensorial perception provides a fundamental point of connection with architectureand points to the essence of the relationship between science and arts. In classicalgeometry and arithmetic, numbers and magnitudes were tangible objects that couldbe shaped, modeled, and transformed from linear to planar and to solid. Theycould be drawn, measured, and compared. Among the main research tools thatancient mathematicians used were the figurate (meaning graphic or figurative)representation of quantities, the visual comparison of magnitudes, and the theoryof mean proportions. In classical antiquity mathematicians utilized a sensorialapproach to scientific research, while artists and architects worked out intellectualdesign procedures relying on numbers and proportional systems to size and shapetheir works. A common concern of all of these disciplines was understandingand imitating the beauty of divine creation. As such, mathematicians, artists, andarchitects potentially shared similar methods and motives.
However, historians of Greek mathematics have a rich bibliography of extantancient treatises on which to base their studies, many more than historians of Greekarchitecture. A cursory comparison between the number of preserved treatises ofmathematics and those of architecture reveals a disparity of number heavily infavor of the mathematical treatises. From ancient Greece, not only have the booksof Euclid (fl. 300 B.C.) come down to us but also those written by Aristotle(384–382 B.C.), Archimedes of Syracuse (287–212 B.C.), Aristarchus of Samos(?–230 B.C.), and Apollonius of Perga (240–? B.C.), among many others. Whileof those regarding architecture, we only have the treatise entitled The Ten Bookson Architecture, a “second-hand source” written much later, in the Roman era, by
Classical Greek and Roman Architecture: Mathematical Theories and Concepts 3
the architect-engineer Marcus Vitruvius Pollio (c. 80–c. 15 B.C.). Being the onlyliterary source dealing with ancient architecture, Vitruvius’ text has been studied andanalyzed at length by scholars (historians and architects). While this text effectivelyclarifies the concerns of Greek architects, while they were designing, in order tounderstand how these concerns were linked to mathematics, we have to turn to themathematical treatises of the era and the sensory methods they employed. For anexample of how important these were, Plato (428–348 B.C.) refers to the role ofvisualization in the development of cognitive process. For Plato, drawn figures aresimply imperfect, although necessary illustrations of a perfect idea: an intermediatestep in thoughts that have to remain purely intellectual.
. . . they further make use of the visible forms and talk about them, though they are notthinking of them but of those things of which they are a likeness, pursuing their inquiry forthe sake of the square as such and the diagonal as such, and not for the sake of the image ofit which they draw . . . And so in all cases. (Plato 2013. Republic, V- 510d)
Plato’s perfect idea may be found in mathematics, but its capacity to beunderstood relies on visualization of its properties. Thus, the connections betweenarchitecture and mathematics in the Classical Greco-Roman world were often foundin particular processes or concepts. Three of these are the focus of the presentchapter. They are the figurate representation of quantities, the visual comparisonof magnitudes, and the theory of mean proportions.
The architects, while defining their project and their design options, go throughall these three procedures themselves. They manipulate geometric figures that arefigurate representation of quantities assigned by the commissioner, or other factors,such as maximum available area, maximum height, maximum cost, etc. Theyvisually compare and evaluate the proportions of the designed object. The controlof the proportions (that modern architects address in a very loose way) was one ofthe main concerns in classical antiquity.
In this chapter each of the three approaches are introduced in terms of theirhistoric mathematical purpose, before the penultimate section of the chapter looksat how they began to have an impact in architecture.
The Figurate Representation of Quantities
Arithmetic
In Plato’s day, mathematicians make a clear and fundamental distinction betweenarithmetic and logistic. “Logistic” deals with numbered things, rather than numbers.It is the art of calculation: it comprises the ordinary operations of adding, subtract-ing, multiplying, and dividing. “Arithmetic” is the science that considers numbersin themselves (Heath 1981). Classical arithmetic stems from Pythagoras’ (c. 580–c.495 B.C.) “Theory of Numbers” which claims that not only do all things possessnumbers, but all things are numbers. Discussing Pythagorean principles, Aristotle(one of the earliest commentators) explains:
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Fig. 1 Polygonal numbers
Fig. 2 Adding triangularnumbers
. . . since it seemed clear that all other things have their whole nature modeled uponnumbers, and that numbers are the ultimate things in the whole physical universe, theyassumed the elements of numbers to be the elements of everything, and the whole universeto be a proportion or number. (Aristotle 1989, Metaphysics, A.5, 985 b 27-986 a 2)
Aristotle also relates how Pythagoreans represented numbers by arranging peb-bles according to various geometric patterns. The disposition of the pebbles (visiblepoints), one for each unit of the number, gave figurative shapes to the quantities. Thequantities were thus grouped together according to their common fitting shapes, andtherefore numbers were called “triangular,” “square,” “pentagonal,” “hexagonal,”and so on. Moreover, the numbers in each category had peculiar proprieties that thegraphic representation itself brought to light and sufficed to demonstrate (Fig. 1).
The manipulation and transformation of polygonal quantities also reveal arith-metical proprieties and relationships between numbers, simply deductible by visu-ally observing the graphic patterns and shapes achieved by varying the arrangementof the pebbles. For instance, the sum of two successive triangular numbers givesa square number, whereas the duplication of a single one generates an oblongnumber which is a rectangular number whose sides are in superpartiens proportion(n/n + 1). While square numbers are found by summing the successive odd termsof the natural series, oblong numbers are found by summing the successive evennatural numbers (Fig. 2).
The table of the first ten terms of the various polygonal series shows that theunit, 1, is common to them all. Conversely, the double, 2, is not a regular polygonalnumber and does not appear in any series. The second term of each series equals thenumber of sides of the generating polygon which depicts the series. Some quantities
Classical Greek and Roman Architecture: Mathematical Theories and Concepts 5
Table 1 The first ten terms of the various polygonal series
1 1 1 1 1 1 1
2 3 4 5 6 7 8
3 6 9 12 15 18 21
4 10 16 22 28 34 40
5 15 25 35 45 55 65
6 21 36 51 66 81 96
7 28 49 70 91 112 133
8 36 64 92 120 148 176
9 45 81 117 153 189 225
10 55 100 145 190 235 280
may also have several different shapes. For example, 55 is either the tenth triangularnumber or the fifth heptagonal number. The series of hexagonal numbers can befound by picking every other term of the triangular series starting from 1, whichmeans that every hexagonal number is also a triangular number. Along a singlecolumn of the table, the difference between two terms is constant, and this constantis equal to the first triangular number of the preceding column. Thus, the successivepolygonal numbers of the same row form an arithmetical progression whose intervalis equal to the first preceding triangle (Table 1).
Further playful manipulations and experiments led the arithmetician to developthree-dimensional representations. The game, consisting of piling up the pebbles inpyramidal shapes, leads from planar to solid arithmetic. The series of pyramidalnumbers derived from various polygonal bases are revealed by arranging thedecreasing terms of a single polygonal series in a steady assembly.
Triangular pyramidal numbers are obtained by summing the successive triangularnumbers: 1, 1 + 3 = 4, 1 + 3 + 6 = 10, 1 + 3 + 6 + 10 = 20, etc. Squarepyramidal numbers come from the summing of the square numbers: 1, 1 + 4 = 5,1 + 4 + 9 = 14, 1 + 4 + 9 + 16 = 30, etc. The same procedure generates the seriesof numbers shaped like a pentagonal, hexagonal, heptagonal, or n-gonal pyramid.The solid numbers of each category are obtained by summing the planar numbersof the relevant polygon (Fig. 3).
The properties that tie pyramidal numbers together match the relationshipsexisting between polygonal numbers, and they can be observed in the table of thefirst ten terms of the various pyramidal series. The unit, 1, is common to all series
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Fig. 3 Triangular and pentagonal pyramidal numbers
Table 2 The first ten terms of the various pyramidal series
1 1 1 1 1 1 1
2 4 5 6 7 8 9
3 10 14 18 22 26 30
4 20 30 40 50 60 70
5 35 55 75 95 115 135
6 56 91 126 161 196 231
7 84 140 196 252 308 364
8 120 204 288 372 456 540
9 165 285 405 525 645 765
10 220 385 550 715 880 1045
and its shape belongs to any kind of pyramid. The double and the triple, 2 and 3, haveno pyramidal shape. The second term of each series indicates the number of facesof the pyramidal solid that depicts the series. Some planar polygonal numbers showalso among the pyramidal numbers. For instance, 55, which is both a triangular and aheptagonal number, is also a square pyramidal number of five levels. Some numberscan take more than one pyramidal shape: 196 is either a pentagonal pyramid of sevenlevels or a heptagonal pyramid of six levels. 196 is also a planar square number withside 14.
Classical Greek and Roman Architecture: Mathematical Theories and Concepts 7
From this quick overview, it is clear that some numbers, but not all of them, canswitch from planar to solid and of course linear.
The sum of two successive tetrahedrons is a square pyramidal number, and –more generally – along a single column of the table, the difference between twosuccessive numbers is a constant equivalent to the first tetrahedron number of thepreceding column. This implies that the series formed by the various successivepyramidal numbers of the same amount of levels is a simple arithmetic progression(Table 2).
Geometry
The Pythagorean representational system applies only to a few arithmetical quanti-ties: those that can fit in this particular geometric bidimensional or tridimensionalshaping. In contrast, geometry is a non-arithmetized discipline which uses a moreflexible representational system, in which any quantity can assume either a mono-, bi-, or tridimensional shape and can therefore be modeled according to variousfigures. Numbers no longer belong to separate definite categories but can betransformed into whatever shape is most convenient for the purposes of specificproblem-solving. They are no longer natural numbers, or integers, but “quantities”or “magnitudes”: forms that can be either larger or smaller and placed together or inopposition to one another.
Classical geometry is said to have originated in ancient Egypt with the necessityof measuring land. However, ancient Greek mathematicians made a clear distinctionbetween geometry and geodesy: similar to the division between arithmetic andlogistic. “Geodesy” is the art of mensuration, not confined to land-measuring butcovering generally the practical measurement of surfaces and volumes (Heath1981). Geometry is a theoretical science. In geometry, quantities are representedeither by lines, surfaces, or volumes, and arithmetical operations are graphicallyequivalent to drawing figures. When numbers are lines, addition means lengtheningan initial line; multiplication between two numbers means drawing a rectangleor constructing a parallelepiped when three numbers are involved. Therefore,multiplying a number by itself creates a square and multiplying it once again createsa cube.
We divided all number into two classes. The one, the numbers which can be formed bymultiplying equal factors, we represented by the shape of the square and called square orequilateral numbers. . . . The numbers between these, such as three and five and all numberswhich cannot be formed by multiplying equal factors, but only by multiplying a greaterby a less or a less by a greater, and are therefore always contained in unequal sides, werepresented by the shape of the oblong rectangle and called oblong numbers. (Plato 1967,Theaetetus, 147e – 148ab)
Plato is not explicit when discussing volumes and thus different shapes canbe created according to the kind of arithmetical operation. When three differentnumbers are multiplied, the resulting parallelepiped will be scalene; if the threenumbers are equal, a cube will be constructed; and if two of the three numbers are
8 S. Duvernoy
equal and different from the third, a square-base prism is generated. The shape ofthe solid reflects the relationship between the numbers involved in its construction.
The geometric representation of numbers is unlimited and can be applied to anyquantity. This is the abstract intellectual tool that allows the scientist to switch frompractice to theory and to draw general theoretical conclusions from the study ofsingle examples. In ancient geometric texts, alphabetical symbols are associatedwith these graphics; however, they must not be confused with the arithmeticalnotation system. These symbols name the objects, indicating either points suchas the vertices of the figures or the quantities themselves represented by linearsegments or surfaces. The written demonstrations accompanying the graphics canthus refer to the letters, stating relationships of parity or inequality between them,heralding the later algebraic equation system of calculation (Fig. 4).
The Visual Comparison of Quantities
The graphic representation of mathematical variables using lines or other geometri-cal figures makes it possible to visually materialize the so-called incommensurablequantities that is irrational numbers, as well as natural integers. In this way theirrational numbers can thus be manipulated together with rational numbers inresearch processes. The scientific value of the graphic depiction emerges in thisexample in its strongest form, drawing being the only means capable of impartingconcrete form to irrational numbers and thereby proving their existence and givingshape and reality to quantities which would otherwise remain “invisible.”
The more ancient geometric problem regarding incommensurability is theduplication of the square and the proportion between its side and its diagonal.Socrates’ lesson to the slave about this predicament, related by Plato in the Menon(82a-85b), is the quintessential illustration of the intellectual power of the drawingin a process of rational deduction using the simple means of visual perceptionand comparison of figurate quantities. In this example, Socrates inquires into thenatural virtues of man, and in so doing he demonstrates the potential of graphicrepresentation in the development of knowledge and science. The figure illustratingthe graphic solution to the duplication of the square, a visual image, or sensibleobject allows anyone to become aware of the intelligible relationships betweenopposing quantities and to draw his or her own conclusions regarding the obviousevidence. Senses and sensorial perception are common to mankind. The capacityto observe and understand is innate and latent in anyone and does not come froma cultural privilege or a high level of education. In order to learn and progress inconscious knowledge, it is sufficient to exercise one’s natural skills. The educationof the neophyte or the methodology of the scientist need only concentrate on howto observe.
Socrates, in showing the figure, does not give the conclusion, because to see the figure, as amathematical figure, actually means knowing how to look at it, how to read it, in short howto think it. (Caveing 1996)
Classical Greek and Roman Architecture: Mathematical Theories and Concepts 9
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10 S. Duvernoy
It is notable that most of the propositions that Euclid includes in the 13 books ofthe Elements are demonstrated through graphics, whose interpretation requires thevisual comparison of figurate quantities. The accompanying text to each diagramguides the learner in the sensitive reading of the scheme, in the tangible evaluationof the entities, and in the appraisal of their perceptible equalities, complementarities,or differences.
A primary property of the visual comparison between figurate quantities is that itprovides the only means of perceiving the value of the irrational magnitudes. Theseentities can be estimated only by being represented together with a known quantity,drawn in the same scale. Thus, in the realm of geometry, a quantity in itself cannotbe considered “incommensurable,” in the sense of not equal to a natural integer, butit is its ratio to another quantity – representing for instance the unit – that may beincommensurable. Only ratios between quantities may be nonmeasurable and non-computable. In book X, Euclid defines irrationality in terms of incommensurabilitywith a given line. The concept of rationality – or irrationality – refers to the qualityof a magnitude. The notion of commensurability – or incommensurability – refers tothe quality of the ratio between two magnitudes. Two quantities are commensurableif they are linked by some plain relationship, the simplest one being the fact ofhaving a ratio equal to some ratio existing between two natural integers.
Ancient mathematicians discerned several kinds of commensurability. Platoalready introduces two of them: commensurability of lines and commensurabilityof surfaces, since the commensurability of two surfaces obviously implies theexistence of a particular kind of relationship between the respective sides of thesesurfaces.
All the lines which form the four sides of the equilateral or square numbers we calledlengths, and those which form the oblong numbers we called surds, because they are notcommensurable with the others in length, but only in the areas of the planes which they havethe power to form. And similarly in the case of the solids. (Plato 1967, Theaetetus, 148b)
In order to illustrate the two different kinds of commensurability mentioned here,it is sufficient to go back to the problem of the duplication of the square. The unit,1, and the square root of 2 are two incommensurable lengths, whereas their squares,1 and 2, are commensurable, being in plain ratio from simple to double. Therefore1 and square root 2 are two quantities commensurable-in-square-only.
Book X is by far the longest book of The Elements by Euclid: it contains 117propositions, while the second longest, Book I, contains only 48. It is also the mostdifficult to understand since it is entirely dedicated to the study of the incommensu-rability concept in terms that are no longer familiar to modern readers. In this workEuclid sets out 13 different kinds of irrationality and commensurability, startingby providing precise definitions of the notions mentioned by Plato. Quantities maybe commensurable-in-length or commensurable according to the squares that canbe constructed on their sides. Magnitudes commensurable in length are those thathave a common divisor (which may be either a natural integer or an irrationalnumber). Magnitudes commensurable-in-square-only are those whose squares areboth multiples of a common quantity. One type of relationship does not categorically
Classical Greek and Roman Architecture: Mathematical Theories and Concepts 11
exclude the other: the same quantities may be both commensurable in length and insquare.
Rectangles contained by lines commensurable-in-square-only are medial quan-tities (areas), and therefore the side of the square equal to this rectangle is amedial quantity (line) too. Since a rectangle is the mean proportional between thesquares on its sides, medial lines are the mean proportional between two linescommensurable in square only. Medial areas can be arithmetically interpreted asthe square root of a non-square rational number, and medial lines are the fourth rootof the same number. A magnitude equal to the sum of two lines commensurable insquare only is a binomial line. A magnitude equal to their difference is an apotomeand so on.
Among the 13 possible kinds of irrationality listed by Euclid in book X, only oneis defined as being the mean proportional between two magnitudes commensurablein square only (the medial); the others are defined by adding or subtractingincommensurable quantities, either areas or lines, leading to one subclassificationfor medial magnitudes and another one for binomials.
All these nuances disappear in the realm of modern mathematics, where algebraand decimal system offer a new approach to the study of irrationality and commen-surability. Moreover, contemporary arithmetized geometry erases the true figurateshapes of the quantities studied in book X, where “medial” or “binomial” may be thecharacteristic of either a line or a surface, generating several further proprieties ineach case. In order to understand the reason and the sense of Euclid’s investigations,we have to understand the ancient context of non-arithmetized geometry, in whichthe manipulation of shapes and figures predominates, and a far from negligibleaspect of which is the study of regular polygons and their inscription in a samecircle. This is the first step to the study of the regular “Platonic” polyhedra inscribedin a single sphere that was to be discussed in book XIII of the Elements. The purposeis to find some qualitative as well as quantitative relationship between the sidesand the diagonals of an n-gon or some qualitative definition of the ratio of thecircumdiameter to a side, a diagonal, etc.
The Theory of Proportion and Means
It is commonly accepted that the theory of proportion and means appeared inthe history of mathematics simultaneously with the concept of irrationality, thePythagoreans’ discovery of the incommensurability of the diagonal of the squareto its side, and later on, the incommensurability of the edges of two cubes whosevolumes are in a ratio from simple to double. It is important to distinguish betweenthe concept of a ratio and the concept of proportion. They refer to different kindsof mathematical notions and are not interchangeable. A ratio is the relationship (afunction) linking two quantities: it may be either commensurable or not, accordingto the various distinctions stated by Euclid. He himself defines a ratio in these terms:
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.(Euclid 1908. Book V, Definition 3)
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In contrast, a proportion is the relationship (a condition) linking three or morenumbers or quantities. For instance, if the ratios of two pairs of magnitudes areequal, then the three or four magnitudes are proportional. In ancient Greece theword mean applies either to a sequence of three terms in continuous proportion orto the middle term which ties together the two extremes.
Ratios, proportions, and means are calculation tools both in arithmetic andin geometry. Ratios between natural integers belong of course to the realm ofarithmetic, but proportions and means are calculation systems that apply to bothdisciplines, being the only possible scientific and computational support of thevisual and sensitive comparison of incommensurable figurate quantities that are soabundant in the realm of geometry.
The three different kinds of means reported by Plato in his dialogues are thearithmetical, geometric, and harmonic means. In the arithmetic mean, the secondterm exceeds the first by the same amount as the third exceeds the second. Inother words, the sequence of numbers is formed by successively adding a constantquantity to each one, thus creating regular intervals of a constant length between thenumbers, and each term is equal to half the sum of its predecessor and its follower.If a and b are the two extremes, X, the midterm − the arithmetic mean – can bewritten as follows: X= (a + b)/2. In the geometric mean, the second term is to thefirst as the third is to the second: the ratio is constant. The sequence is found byconsecutively multiplying the terms by the same amount. Therefore each term isthe square root of the product of its predecessor and follower. The midterm of atrio – the geometric mean – can be written as follows: X = √
ab. In the harmonicmean, three consecutive terms are such that by whatever part of the first, the secondexceeds the first; the third exceeds the second by the same part of itself. In modernnotation, a and b being the extremes, X – the harmonic mean – is found by theequation: X = 2ab/(a + b).
Several other means were added later by other mathematicians: three of thembeing the subcontraries of the geometric and harmonic, but none of them were asimportant for scientific progress, most of them being the result of purely systematiccomputational investigation.
Musical Proportions
The most ancient discovery to use the theory of proportions is the definition ofmusical harmony proposed by the Pythagoreans. The sensorial acuity behind theirscientific research was, in this case, no longer visual but auditory perception.
Faithful to the basic concept found in the Theory of Numbers, which statesthat all things are numbers, the question for Pythagoreans was to find a way tonumerically quantify the sounds that musicians play on their flutes and lyres. Thus,which numbers are capable of describing musical notes? The desire to be able tocompose such notes in a thoughtful and melodic sequence suggests a reflection ontheir intrinsic properties and reciprocal relations.
Classical Greek and Roman Architecture: Mathematical Theories and Concepts 13
Sound is the sensible effect of a physical phenomenon: vibrations that travelthrough the air. It is a function of the frequency of that vibration which in turndepends on the length of the string of the lyre to be plucked by a musician or thelength of the flute through which air is blown.
If the longest string of a tetrachord, whose length is double the shortest one,produces a sound that is an octave higher than the small one, of what length must thetwo middle chords be in order to produce the intermediate notes, being the middlesounds that the ear can perceive? How can intervals between 1 and 2 be defined? 1and 2 being two successive numbers?
Plato reports the procedure for solving this by having Socrates speak words thatsound slightly ironic.
. . . they talk of something they call minims and, laying their ears alongside, as if trying tocatch a voice from next door, some affirm that they can hear a note between and that this isthe least interval and the unit of the measurement, while others insist that the strings nowrender identical sounds, both preferring their ears to their minds. (Plato 1935, Republic, VII,531a-c)
The Pythagoreans had previously demonstrated that there is no rational geo-metric mean between two numbers in superpartiens ratio that is between twoconsecutive integers (n, n + 1). Therefore any answer will produce irrationalquantities. The interval of the octave has to be “filled” with the two others means:arithmetical and harmonic. However in order to transform the four magnitudes inround numbers, it is necessary to give to the extremes of the octave the values of 6and 12, rather than 1 and 2. Thus the arithmetic mean is 9 and the harmonic is 8.
This formulation generates a sequence of natural integers: 6, 8, 9, and 12, and theinterval of the octave – or diapason – has been filled. The lengths of the two middlechords of the lyre have also been determined, and they are commensurable with theshortest and longest ones. Several intermediate intervals have also been created. Theinterval between 6 and 9, equal to the interval between 8 and 12 (whose length is1 + 1/2), is the fifth – diapente – and its extremes are in a ratio of 2:3. The intervals6–8 and 9–12 whose length is 1 + 1/3 are the fourths – or diatessaron – and theextremes are in a ratio of 3:4. The last interval, between 8 and 9, whose length is1 + 1/8, is the tone. The fifth is equal to three tones and a lemma; the fourth is twotones and a lemma. The Pythagorean quantification of the lemma is 256/243. Themusical intervals and their subdivisions appear clearly on the piano keyboard, whichshows the repetition of the successive octaves, and the various notes of each, betterthan any other musical instrument (Fig. 5).
In the dialogue Timaeus, Plato applies the Pythagorean theory of musicalproportions to the explanation of the creation of the soul of the world, while thecreation of the body of the world was made possible thanks to the geometrical mean.The mathematical process is both the guarantee and the explanation of the beauty ofthe divine composition. In Ancient Greece, mathematics, metaphysics, philosophy,and artistic creation were linked together by a common sense of awe about thebeauty of the universe and the divine creation and by the desire to understand,explain, quantify, and imitate divine beauty.
14 S. Duvernoy
Fig. 5 The Pythagoreanmusical proportions
The Duplication of the Cube
With parallels to the solution developed for musical intervals, the problem of theduplication of the cube was solved applying the system of proportion and means.The duplication of the cube, the trisection of any angle, and the squaring of thecircle are the three classic problems of ancient mathematics that triggered scientificresearch for centuries and for which various solutions were put forward. All threeseek a precise quantification of an irrational quantity and its commensurability witha known rational magnitude. It is now well known that the squaring of the circle hadno solution. The trisection of any angle was the easiest to solve. The duplication ofthe cube can be solved in many ways, some more complicated and others less.
Hippocrates of Chios (470–410 B.C.) was the first to take an initial step towardthe solution of the duplication of the cube. We are told by Proclus Diadochus (412–485 A.D.) that he “reduced” the problem to the need to find two mean proportionals,in continued proportion, between two given straight lines.
In modern mathematics the problem can be reduced to the finding of anapproximate value for the cubic root of 2. But in Hippocrates’s time, when theirrational quantities only had a graphic shape appearing on a geometric figure,the arithmetical problem of finding two geometric means between two numbers,notably – once again – between 1 and 2 (where 3√2 lies), becomes a drawingproblem involving finding two mean proportionals between two straight lines. Wemay only guess what kind of reasoning Hippocrates pursued in order to reach to his
Classical Greek and Roman Architecture: Mathematical Theories and Concepts 15
Fig. 6 The geometrical solution to the doubling of the cube attributed to Apollonius
conclusion. We may suppose that he extended the methods of calculation typical oflinear and planar geometry, to solid geometry by simple analogy. If the geometricmean of two quantities is the square root, that is, a line equal to the side of thesquare constructed from the product of the two extremes, then a cubic root too isto be found in a geometric progression of linear quantities. Proposition 17 in bookVI of the Elements shows an exact bidimensional correspondence to the duplicationof the cube as “reduced” by Hippocrates a century and a half earlier. His assertion,which was not yet a solution, had a massive influence on the later mathematicianswho continued to work on the problem, taking it as a starting point. “Reducing” aproblem does not mean solving it; it simply means making it conform to a familiarpattern of queries. Thus the work of Hippocrates is a precious indication that thetheory of proportion and means was already largely in use as a method for problem-solving in his day, and he only expanded it from the realm of planar geometry to thefield of solid geometry (Fig. 6).
Art and Architecture
While mathematicians approached their work with an artistic sensibility that pushedthem to unveil the beauty inherent in the laws of nature, artists and architectsattempted to identify a mathematical support to their art.
The sculptor Polycleitus (mid-fifth century B.C., roughly contemporary toHippocrates of Chios) is famous for his accurate study of the proportions of thehuman body. That is, the commensurability of the various measures of the individualparts of the body and their commensurability to the whole. His studies resulted inboth a theoretical treaty “the Canon” and a model statue, the “Doryphoros.” Theoriginal statue is lost but some Roman copies have been preserved. The treatise
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is unfortunately lost too, but some quotations are found in later texts includingthe writings of the Greek physician Galen (129–c.200 A.D.), a scientist obviouslyinterested in the secrets of human anatomy. Modern scholars have attempted atreconstructing what was written in the “Canon.” It seems that Polycleitus unveileda proportional rule, starting from the lengths of the finger phalanges, the ratio ofthe finger length in respect to the hand itself, the hand to the forearm, and soon, showing that all those dimensions are in continuous geometric proportion. Hethus established a mathematical basis on which artists could rely to control themaking of well-proportioned statues. This mathematical analysis of the beauty ofthe human body had a strong and long-lasting impact on artists and architects sinceit tended to prove that divine beauty (Man was created by God) was indeed basedon proportional rules.
Another ancient claim for the mathematical rules of the visual arts is reported byPliny the Elder (23–79 A.D.) in his Naturalis Historia, Book 35. Pliny tells us aboutthe Greek art school which was founded in Sicyon by the painter Eupompus (fourthcentury B.C.). The school flourished under the guidance of Eupompus’s successorPamphilus (fourth century B.C.), who is described by Pliny as the first painter with atheoretical background in arithmetic and geometry. Pamphilus claimed that art couldnot attain perfection without the support of mathematical knowledge. Consequentlyhe elevated visual arts to the level of liberal arts, to such a level that free-bornchildren could be given drawing lessons.
Similarly to painting and sculpting, the Greek architectural search for beautyand harmonious visual effect expresses itself through numbers and arithmeticalratios. Numbers, ratios, and mean proportions are borrowed from, and shared with,other scientific or artistic disciplines where they have proved to be successful,including anatomy and music. Vitruvius provides an extensive literary source aboutthis research on numbers and ratios, but more than the results that he reports, it is themethodology of design, perceptible from his writing, veiled by the compilation ofthe list of rules that is significant. Speaking about temple design, the most importanttypology of Greek architecture, he writes:
The design of a temple depends on symmetry, the principles of which must be most carefullyobserved by the architect. They are due to proportion, in Greek αναλoγια. Proportion is acorrespondence among the measures of the members of an entire work, and on the wholeto a certain part selected as standard. Without symmetry and proportion there can be noprinciples in the design of any temple; that is, if there is no precise relation between itsmembers, as in the case of those of a well shaped man. (Vitruvius 1914. III, 1)
For Vitruvius, the concept of “symmetry” is not the same as its modernmathematical meaning. In Greek and Roman times, the architectural concept of“symmetry” is the equivalent of the mathematical concept of “commensurability.”All dimensions of a building must be commensurable between them. Architectsmust learn from Polycleitus who unveiled the “symmetry” of the human body –God’s creation – and apply the same concept in their design in order to try to attainperfect beauty.
Classical Greek and Roman Architecture: Mathematical Theories and Concepts 17
Conclusion
Relationships between architecture and mathematics are too often considered to befounded on a hierarchy and chronology which put mathematics first and architecturesecond. Such a hierarchy suggests that mathematics existed before art and architec-ture, and the former always provided the latter with a solid scientific background,a sort of database of shapes, geometric figures, numbers, and proportional systems,among which designers could pick at any given time the solution that best answeredtheir needs.
In a time in which borders between sciences, philosophy, and arts were looseand all fields of knowledge and culture were intertwined, mathematics grewsimultaneously with the arts, acting as a catalyst as well as a recipient of progressand cultural evolution. The practical aspects and problems related to architecturaldesign and building operation, often pushed to the development and progress oftheoretical research and surely contributed to the arithmetization of geometry.
As it often happens in Greek mythology, the classic problem of the duplication ofthe cube is coupled to a legend, the famous “legend of Delos,” and is therefore oftenreferred to as the “Delian problem.” The legend of Delos is the most ancient and besthistorical example of interactive relationship between architecture and mathematics.The legend says that the people of Delos, struck by a severe plague, asked the oracleof Apollo how to calm the Gods’ wrath. The answer was that they only had to builda new altar to Apollo, twice as big as the existing one, which was of a cubic form. Sothe locals immediately obeyed the divine wish and built a new altar, whose edgeswere twice as long as the previous one . . . but the plague did not stop. In fact bydoing so, the altar had been multiplied by eight instead of two.
The legend of Delos is reported in every essay on the history of ancientmathematics, with some variation from book to book, and with more or lessdetails, but the essence of the problem is always the same. Tradition and mythologyassociate this famous classic mathematical problem with an architectural problem.The goal is to build a votive monument of a given shape, cubic, and dimension.In order to determine the dimension, architects turn to mathematicians, who areunable to give a precise numerical value. Contemporary knowledge in arithmeticstill lacks some fundamental answers to ordinary and practical problems in the fieldof construction. With that single and basic query, architectural design promotedefforts in theoretical mathematical research which would last several centuries.
Independently of the fact that the legend might be true or not, it illustrates veryexplicitly the quality of the relationship between mathematics and architecture inancient Greece, and more generally between theory and practice, which are notalways linked by a mere cause-effect sequence, but often interfere in reciprocalinfluence and stimulation.
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Cross-References
�Classical Greek and Roman Architecture: Examples and Typologies
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