The first, "Proportions in ancient Egyptian architecture," is where the author critically reviews the theories put forward to explain proportions in Egyptian architecture from the nineteenth century up to our time, by authors ranging from Viollet-le-Duc to the Egyptian architect and Egyptologist Alexander Badawy.
This Rossi calls the 'architectural' approach, and it is in this part that she displays for the reader the never-never land of geometrical lines superimposed on plans and sections in search of neat proportional relations among component parts of architectural entities. One point of great interest here is a distinction drawn between "two main groups of scholars who have dealt with ancient Egyptian architecture: architectural historians and Egyptologists" p.
To this end, in order to brush away the misconceptions and "faulty assumptions" generated by searching for mathematical or proportional rules in Egyptian architecture on the basis of modern plans and sections, Part II provides the reader with a number of examples of Egyptian architectural working drawings and models along with their assessment, although these by no means suffice to give one a definitive sense of the authentic Egyptian understanding and practice of architectural design and building either.
What shines through most clearly in the book is perhaps this mnemonic nature of Egyptian architectural design and building praxis rather than the presence of a tangible Egyptian mathematical system which the architects drew on in their designs of monuments.
The aim of Part III, entitled "The geometry of pyramids," is nevertheless stated by the author to be an attempt to reconcile the 'architectural' and 'Egyptological' views, and this Rossi mainly carries out through combining the study of ratio and proportion with a well-documented and straightforward Egyptian manner of expressing and managing the slopes of a variety of pyramids from all ancient Egyptian history. There are a number of topics and arguments in the book that are of particular interest and at which one could take a closer look. Part I introduces two mathematical phenomena that have concerned scholars in relation to Egyptian architecture: three types of triangles and the Golden Section.
The triangles are the well-known , the equilateral, and the isosceles triangle with the ratio between the base and the height, and hence called the triangle by Badawy, and 'Egyptian' by Viollet-le-Duc, who had erroneously identified it as the vertical section of the pyramid of Khufu.
The merit of this ratio, according to Viollet-le-Duc, is its not being derived from either the half or the third or the fourth of the larger dimension, and hence its encompassing a relation which the eye could not define, resulting in "a means of obtaining the contrasts which are necessary for satisfying the first law of proportions" p.
Certainly analogous in this regard is the Golden Section, not easy to define in such terms either, which Rossi deals with in some detail. She first introduces the rules and mechanics of this ratio with diagrams, and draws attention to the fact that its numerical value, 1.
Rossi is rather skeptical about theories that have tried to see the Golden Section in Egyptian architectural design as the preferred ratio, even though she does acknowledge that it was one of the proportions used by Egyptian architects, along with the proportions of the triangles already mentioned. Even though the author seems to have made good use of nineteenth-century primary sources that deal with such theories in her research, their presentation in the book is over-synoptic, without clarification for the reader as to what exactly these theories entailed. Furthermore, Rossi herself seems somewhat torn between the existence and the absence of a set of clear rules in Egyptian architectural design, as she mentions the theory of Badawy as "able to explain many factors," a theory which suggests that a number of triangles including the and the Golden Section were used by the Egyptians in the design of their monuments among other geometric forms and ratios.
Despite many points of criticism directed toward Badawy, Rossi acknowledges that "Badawy's schemes seem to work," and goes into a greater detail than she does for others in explaining his theory of how certain triangles seem to have been used in laying out the ground plans of certain Egyptian temples.
Within Part I, the author also addresses the question to what extent the Egyptians made conscious use of the Pythagorean theorem and the numbers phi and pi. Rossi again cautions against retrojecting modern mathematical notions into Egyptian antiquity and argues that, even though the Egyptians may have used some Pythagorean triplets, "this does not imply that the Old Kingdom architects were the creators of the Pythagorean symbolism of numbers or were aware of all the mathematical implications of Euclid's formulation of the theorem of Pythagoras" p.
Even though Rossi's emphasis on eschewing anachronisms in dealing with the Egyptian case is well made, both the theorem of Pythagoras and the numbers phi and pi are such fundamental mathematical concepts that I wonder if it is necessary to discuss, even in the absence of contemporary sources and evidence, whether or not the Egyptians or any other pre-Hellenic ancient people in the area would have been fully familiar with such 'universals' and their implications. Perhaps the fundamental error is to take the theorem of Pythagoras as the 'canonical' or the 'most developed' version of this mathematical phenomenon, as if a complete understanding of the latter could not have been possible without reference to Pythagoras.
Rossi actually comes back to this question in Part III, where she again states that notwithstanding the Egyptians' knowledge of "some triplets, But for multiplication they introduced a method of successive doubling.
For example, to multiply 28 by 11, one constructs a table of multiples of 28 like the following:. The several entries in the first column that together sum to 11 i. To divide by 28, the Egyptians applied the same procedure in reverse.
Using the same table as in the multiplication problem, one can see that 8 produces the largest multiple of 28 that is less then for the entry at 16 is already , and 8 is checked off. The process is then repeated, this time for the remainder 84 obtained by subtracting the entry at 8 from the original number This, however, is already smaller than the entry at 4, which consequently is ignored, but it is greater than the entry at 2 56 , which is then checked off.
The process is repeated again for the remainder obtained by subtracting 56 from the previous remainder of 84, or 28, which also happens to exactly equal the entry at 1 and which is then checked off. In most cases, of course, there is a remainder that is less than the divisor. For larger numbers this procedure can be improved by considering multiples of one of the factors by 10, 20,…or even by higher orders of magnitude , 1,,… , as necessary in the Egyptian decimal notation, these multiples are easy to work out.
Even if the ancient Egyptians would have been flattered by this attitude, the results of this kind of speculation have, unfortunately, little to do with the actual historical and archaeological remains. See also Richard J. The contents of the room objects, people, activi- ties, but also architectural elements like doors and columns were better represented from a frontal point of view, in superimposed registers if necessary fig. Search the history of over billion web pages on the Internet. It may be worth remembering that tt was proved to be irra- tional only in , and to be transcendental in 1 , and that therefore Archimedes himself did not think in terms of it as we do today. Dieter Arnold suggested that this may have been an early project for the interior of the Hawara pyramid, used by the architects to study a first distribution of the rooms.
Thus, one can find the product of 28 by 27 by setting out the multiples of 28 by 1, 2, 4, 8, 10, and Since the entries 1, 2, 4, and 20 add up to 27, one has only to add up the corresponding multiples to find the answer. Computations involving fractions are carried out under the restriction to unit parts that is, fractions that in modern notation are written with 1 as the numerator. A considerable portion of the papyrus texts is devoted to tables to facilitate the finding of such unit-fraction values. These elementary operations are all that one needs for solving the arithmetic problems in the papyri.
The geometric problems in the papyri seek measurements of figures, like rectangles and triangles of given base and height, by means of suitable arithmetic operations. For example, if the diameter is 9, the area is set equal to This is a rather good estimate, being about 0. Overall the book is an easy but dry read.
A very nice feature of the book is that each part begins with an introduction providing an overview of Dr. As far as mathematics in ancient Egypt is concerned, Dr. Rossi refers to several mathematical papyri, including the Rhind Papyrus.
She briefly discusses unit fractions and the doubling and halving methods used to perform multiplication and division. She also elaborates on the ancient Egyptian use of ropes for surveying.
In attempting to find mathematical patterns within the design of the pyramids, she insists upon using the measurements of the era: cubits, palms, and fingers. Unencumbered by modern units of measurement, she is free to use the ancient Egyptian measurement known as the seked to calculate and find patterns among the slopes of all the pyramids in her study.