**Part I**

Proportions in ancient Egyptian architecture

Harmony and proportions in architecture

Throughout the whole history of architecture, the concept of harmony has been the subject of numerous studies and long-lasting discussions.^{1} Harmony may be defined as a correspondence between parts, the result of the composition (or the division) of a whole into consonant parts. Its ancient link with music, where ‘agreeable’ combinations of sounds can be read as mathematical relationships, seems to seal the connection between harmony and mathematics. In art and architecture, however, this correspondence is not easily described. Although it is undeniable that a link between architecture and geometry (and therefore mathematics in general) exists, in different periods the nature of this connection has been identified and judged in different ways.

To Pythagoras and the Pythagoreans is attributed the discovery that tones can be measured in space – that is, that musical consonances correspond to ratios of small whole numbers. If two strings vibrate in the same conditions, the resulting sounds depend on the ratios between their length. The ratio 1:2, for instance, generates a difference of one octave (diapason), the ratio 2:3 produces an interval of one fifth (diapente), and the ratio 3:4 corresponds to a difference of one fourth (diatessaron). The addition of two intervals results in the multiplication of the two numerical ratios, the subtraction corresponds to a division and, therefore, halving an interval equals extracting a square root.

The Pythagorean interest in numbers, as filtered by Plato, generated a tradition that linked philosophy and mathematics in the interpretation of the cosmos. Thus, as Walter Burkert wrote

one is *nous* and *ousia*; two is *doxa*; three is the number of the whole – beginning, middle and end; four is justice – equal times equal – but it is also the form of the *tetraktys*, the ‘whole nature of numbers’ [a ‘perfect triangle’ made up of the numbers 1, 2, 3 and 4]; five is marriage, as the first combination of odd and even, male and female; seven is opportunity (*kairos*) and also Athena, as the ‘virginal’ prime number; ten is the perfect number, which comprehends the whole nature of number and determines the structure of the cosmos.^{2}

During the European Renaissance interest in the theory of proportion in architecture increased considerably. One of the reasons for this was the rediscovery, in 1414 in the Montecassino Abbey, of the treatise *De Architectura libri decem*, written by the Roman architect Vitruvius in the first century BC, and eventually translated from the Latin into the major European languages during the sixteenth and seventeenth centuries. According to some authors, Vitruvius failed to provide a coherent theory of proportions. P. H. Scholfield, however, has explained that the difficulties in the translation of Latin words which appear to have similar meanings, such as ‘symmetria’, ‘eurythmia’, ‘proportio’ and ‘commensus’, generated confusion and misunderstanding among scholars and commentators.^{3} One of the most important elements in Vitruvius’ theory is commensurability: the dimensions of the parts are submultiples of the dimensions of the whole. This seemed to apply especially to the human body, and he suggested that the proportions of a temple ought to be like those of a well-formed human being.^{4}

Vitruvius and the Pythagorean-Platonic philosophy of harmonic numbers were the main source of reference for Renaissance architects. In 1534, the painter Titian, the architect Serlio and the humanist Fortunio Spira all approved the project suggested by the Franciscan monk Francesco Giorgi for the proportions of the church of San Francesco della Vigna in Venice, which had been laid out according to Pythagorean and Platonic theories.^{5} The Italian architects Palladio and Leon Battista Alberti, although with some differences, followed the same principles and based their architectures on simple ratios of small numbers.

The idea of a universal harmony which ruled microcosm and macrocosm began to decline in the seventeenth century, and was completely overthrown in eighteenth-century England. According to David Hume, beauty was not a quality in things themselves, but existed only in the mind of the person who contemplated them,^{6} and Edmund Burke concluded that beauty had nothing to do with calculation and geometry.^{7} Even the connection between architecture and music was heavily criticised and dismissed in favour of a more individual point of view influenced by the limitations of human perception.^{8} At the same time, the sensation that something belonging to the past had been lost started to appear. William Gilpin sadly wrote that ‘the secret is lost. The ancients had it. They well knew the principles of beauty; and had that unerring rule, which in all things adjusted their taste. (. . .) And if we could only discover their *principles of proportions*, we should have the arcanum of the science, and might settle all our disputes about taste with good ease’.^{9}

The idea of a universal harmony was revived in the second half of the nineteenth century, together with research, in art and architecture, into a common rule which could link the past to the present.^{10} The study of Emeric Henszlmann published in 1860, for example, bears a significant title: *Théorie des proportions appliquées dans l’architecture depuis la XIIe dynastie des rois égyptiens jusq’au XVIe siècle*. The author constructed a series of increasing and decreasing ratios between catheti of right-angled triangles and suggested that these values had been used by the architects of different cultures for millennia.^{11} However, in 1863 the French architect and architectural historian Eugène Viollet-le-Duc expressed a different opinion about the nature of harmony. Being a theorist but at the same time a great expert in construction and restoration, he had a more balanced view of the link between abstract mathematics and practical operations. According to him,

going further back and examining the monuments of Ancient Egypt, we also recognize the influence of a harmonic method, but we do not observe the artists of Thebes subjected to a formula; and I confess I should be sorry if the existence of such formulas among artistic peoples could be demonstrated; it would greatly lower them in my estimation; for what becomes of art and the merit of the artist when proportions are reduced to a formulary?^{12}

The results of the search for a unique rule, especially in the cases when they were summarised in a mathematical formula or a geometrical construction, are quite heterogeneous and often contradictory. Some scholars thought that they had found the solution in a geometrical figure: Viollet-le-Duc himself believed that triangles were the basis of every good architecture,^{13} Odilio Wolff favoured the hexagon,^{14} Ernst Mössel the circle,^{15} and Jay Hambidge the so-called ‘root rectangles’, that is, rectangles in which the short side was equal to the unity, and the long side respectively to √2, √3, √4 and √5.^{16} His system, which he referred to as ‘Dynamic Symmetry’, is also related to the most successful among the geometrical constructions evoked by the scholars of the nineteenth and twentienth centuries: the Golden Section.^{17}

This proportion appeared to satisfy all the requests for a proper ‘universal rule’. From a mathematical point of view, it could have a relatively simple geometrical construction, but at the same time an extremely complicated theoretical background, which lent itself very well to a symbolic interpretation. Moreover, at the beginning of the twentieth century several scholars suggested that the structure of many natural forms was based on this proportion.^{18} In 1854 Adolf Zeising claimed that he had discovered that the Golden Section ruled the proportions of the whole human body (height and breadth, front and back), and that the same occurred in music, poetry, religion and, of course, architecture.^{19} In the first half of the twentieth century, Matila Ghyka followed his example and applied the Golden Section to nature and art, explicitly referring to a ‘revival of Pythagorean doctrine in science and art’.^{20}

Ancient Egyptian architecture, in particular the Giza pyramids, quickly became one of the favourite subjects to which such an approach was applied. Unfortunately, the majority of these theories, on Egypt or other ancient cultures, are based on our modern mathematical system, which is not necessarily similar to the ancient ones. Part I is entirely dedicated to this discrepancy, and reflects this in its structure. The first section will focus on the theories suggested to explain the proportions in ancient Egyptian architecture and their mathematical background. In the second section, I will adopt the historically correct mathematical point of view and demonstrate that many of those theories are based upon faulty assumptions.

**1**

In search of ‘the rule’ for ancient Egyptian architecture

**Triangles and other figures**

*Three triangles for ancient Egypt*

Before the diffusion of the Napoleonic *Description de l’Egypte*, the available information on Egyptian architecture was fragmentary and imprecise, and did not necessarily create a good impression on architectural historians, especially those accustomed to the reproductions of Greek art and architecture. The French architectural historian Quatremère de Quincy, for instance, was not exactly a supporter of ancient Egypt. He believed that ancient Egyptian architecture lacked ‘order’, meaning a ‘system of proportions of forms and ornaments’, and that beauty and taste were foreign to it.^{1} In his *Dictionnaire Historique d’Architecture*, he wrote that in Egyptian architecture the large size of the construction, the vastness of the composition and the profuseness of signs and objects were due to a lack of science, a lack of creativity, and a lack of taste, respectively.^{2} He concluded that the best way to mould someone’s taste, to develop a feeling for truth and beauty, was to familiarise them with Greek statuary. But if one wished to prevent this feeling from developing, it would be enough to condemn the person to looking at Egyptian statues.^{3} If one looks at the appalling reproductions of Egyptian monuments that accompanied his texts (for example fig. 1), the temptation to agree with him is very strong.

Ancient Egyptian architecture began to be included in the studies on architectural proportions when more reliable reproductions of the monuments became available. In the nineteenth century, drawings from the *Description* and from the publications of early travellers were the sole sources that the architectural historians could use in their studies. Ancient Egyptian texts were being slowly deciphered and by the

Fig. 1: Early nineteenth-century reproductions of Egyptian monuments (from Quatremère de Quincy, *De l’architecture Egyptienne*, pls. 3 and 10).

end of the century, among the mass of material accumulated by archaeologists and travellers, ancient architectural drawings started to appear. For a long time, however, the only available material for architectural historians was represented by more or less precise surveys of the archaeological remains.

In general, not many Egyptian monuments were actually taken into account in nineteenth-century studies on architectural proportions. The aim of the nineteenth-century scholars generally was just to connect Egypt to the following history of architecture and to make sure that its monuments harmonised with the rest, rather than study them as a separate subject. In 1854 Zeising dedicated just a paragraph to the Egyptian canon of proportion for the human body. In 1860 Henszlmann started his study with a few observations on the Egyptian Twelfth Dynasty, but his main concern was certainly Greek architecture. In 1924, Hambidge wrote that

Saracenic, Mahomedan, Chinese, Japanese, Persian, Hindu, Assyrian, Coptic, Byzantine, and Gothic art analyses show unmistakably the conscious use of plan schemes and all belong to the same type. Greek and Egyptian art analyses show an unmistakable conscious use of plan schemes of another type. There is no question as to the relative merit of the two types. The latter is immeasurably superior to the former. This is made manifest as soon as the two types are tested by nature.^{4}

No practical examples of the application of this theory on actual Egyptian buildings, however, are included in this publication. The first publication to take into account ancient Egyptian textual sources and various archaeological material, and to suggest an interpretation of over fifty Egyptian monuments, was the monograph published in 1965 by Alexander Badawy.

The most successful among the geometrical figures applied to ancient Egyptian architecture (with or without a precise connection with the Golden Section), was the triangle. In particular, three triangles: the 3-4-5, the equilateral and the triangle called ‘Egyptian’ by Viollet-le-Duc and 8:5 by Choisy and Badawy. The first is a right-angled triangle which belongs to a peculiar group of right-angled triangles in which all the three sides correspond to whole numbers. The second is a triangle with three equal sides, while the third is an isosceles triangle (two sides equal and one different) in which the ratio between the base and the height is about 8:5. Originally, as we shall see in the next paragraph, in Viollet-le-Duc’s suggested theory the equilateral and the ‘Egyptian’ or 8:5 triangle were geometrically connected, the latter depending on the former. But this link was soon forgotten, and the supposed use of the second triangle was later interpreted as an approximation of the Golden Section. The story of the evolution of the theories discussed in the following section is a tale of confusion and misunderstanding among scholars and among geometrical figures, which I will attempt to clarify.