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The Geometric Grids of the Hieratic Numeral Signs Mediterranean Arhaeology and Archaeometry, Vol. 12, No 1, pp.129-138 Copyright © 2012 MAA Printed in Greece. All rights reserved. THE GEOMETRIC GRIDS OF THE HIERATIC NUMERAL SIGNS Hossam M. K. Aboulfotouh Faculty of Environmental Design, King Abdulaziz University, KSA Faculty of Fine Arts, Minia University, Egypt Received: 21/12/2011 Accepted: 24/02/2012 Corresponding author:[email protected] & [email protected] ABSTRACT The paper discusses the geometrical designs of the hieratic numeral signs. It shows the regular-grid-patterns of squares upon which, the shapes of the already decoded hieratic numeral-signs, have been designed. Also, it shows the design of some hieratic numeral signs, based on subdividing the circle; and the hieratic signs of modular notation. It might reveal the basic geometrical level of understanding of anonymous ancient Egyptians who designed them some four thousand years ago. KEYWORDS: Egyptian Mathematics, Rhind Papyrus, Moscow Papyrus, Hieratic Notation, Numbers, Modular Notation 130 HossAM M. K. AboulfotouH 1. INTRODUCTION hieroglyphic writing, Georges Ifrah (1998, 171) said "hieratic signs were indeed de - For almost fourteen decades, great efforts rived from hieroglyphs"; however, he have been done by many scholars in order pointed out to that "the relationship be - to interpret and introduce the legacy of the tween hieratic numerals and hieroglyphs is ancient Egyptian mathematics to the scien - difficult to see". Most hieratic numeral signs tific community. The most known early seem follow different architectonic concept, publications were by, August Eisenloher which is not based only on the idea of (1877), Thomas Eric Peet (1923), and Arnold adding characters of pre-assigned values to Buffum Chace et al (1927-1929), for inter - indicate numbers, i.e., the values of one and preting part or all of the hieratic text of the 10 n, where n equals any of the integers from so-called Rhind Mathematical Papyrus 1 to 6 at most. Boyer and Merzbach (2010, (RMP); and also by, W. W. Sturve (1930) for Ch2) expressed their opinion concerning interpreting the hieratic text of the so-called the transformation from the hieroglyphic Moscow Mathematical Papyrus (MMP); numerals into hieratic numerals by saying etc 1. Thanks to the great efforts that have "Numeration remained decimal, but the te - been done by many scholars in this area of dious repetitive principle of hieroglyphic knowledge; particularly their works per - numeration was replaced by the introduc - taining to decoding the meanings of the hi - tion of ciphers or special signs to represent eratic numeral signs, which to some extent digits and multiples of power of ten"; and do not match their equivalents in the hiero - added "the principle of cipherization intro - glyphic writings. The hieroglyphic numeral duced by the Egyptians some 4000 years system is as old as the days of King ago, .. is one of the factors that marks our 2 Narmer ; according to Archibaled (1930, own system in use today". They meant, e.g., 3 109-212) it goes back to 3500 BCE . Gillings writing 26 using only two signs or writing (1972, 4) commented on the hieroglyphic 746 using only three signs. In contrast, sim - numeral signs of the ancient Egyptians, say - ilar to the opinion of Budge concerning the ing "the method used to represent numbers hieratic letters, Clagett (1991, 5) denoted the must have been at least easier than writing writing of the hieratic numerals by the word their phonetically equivalent words". Sixty "abbreviated", which might imply writing two years before the publication of Gillings, in shorthand, but this is not the case in most Budge (1910, 127-128), based on the works hieratic numeral signs. From the artistic of early scholars, showed the discovered point of view, if one practiced the hieratic way of writing numbers in Egyptian hiero - writing, one might notice that its basic-idea glyphs by repeating and adding vertical is based on drawing the outline of the pro - strokes to represent numbers from 1 to 9, file of the hieroglyphic figure following ei - e.g., means 3 and means 4; and then ther clockwise or anti-clockwise directions from 10 upwards they used various signs 4 of motion, or both together, without closing to represent the progressive hierarchy of the loop of the profile, particularly that for numbers based on the power of ten, i.e., an the live-beings in side-view, e.g., the anti - arch ∩ denotes ten, a horn denotes 100, a clockwise hieratic profile of the hawk (see lotus flower denotes 1000, a finger de - the first black letter in RMPp50, in figure-5) notes 10,000, a frog denotes 100,000, and looks like witting the two English letters the ideogram of large denotes 1,000,000. li , in small. If one applied that on the hiero - That is, writing numbers in hieroglyphs glyphic numerals, the results will in most was by adding strokes and figures 5. In line numbers be different from what we read in with the opinion of Budge (1910, 4) concern - the heretic manuscripts. That is, contrary to ing that hieratic writing is abbreviation of the case of the hieratic letters, most hieratic tHE GEoMEtRIC GRIDs of tHE HIERAtIC NuMERAl sIGNs 131 numerals are not abbreviations of the hiero - Geometry first came to be known in Egypt" glyphic numerals, e.g., the hieratic sign of and might also imply that they knew the number 7 is not abbreviation of its hiero - right–angled triangle (3, 4, 5) long before the glyphic sign . The author of this paper, days of Pythagoras 6. therefore, investigated the question that "is there an ancient Egyptian unified concept 2. THE NUMERIC GRID SYSTEM leaded to the design of their hieratic nu - OF SQUARES meral signs? As a result of reviewing the de - Based upon the discovered meanings of sign typology of the hieratic numeral signs, the hieratic signs of numerals by early the author of this paper suggests that hier - scholars 7, and upon reviewing the design of atic notations are to many extent derived each of these numeral signs, figure-1 shows from specific geometrical system, which that the hieratic shapes of integers (from 1 might support the note of Herodotus (440 to 10) and the fractions 2/3, 1/2 and 1/3 were BCE, Book II) on what might was the impli - designed using the grid of a thematic rec - cation of the practice of measuring lands in tangular, its dimensions are 4*3 units. Egypt by saying "from this practice I think Figure-1. The design grids of the hieratic signs of integers from 1 to 10, and the fractions: 2/3, 1/2, 1/3, 1/4, 1/8, 1/16, and 1/32. 132 HossAM M. K. AboulfotouH Designing the shapes of integers seem of one unit in the upper right corner of a were based on that one is a side of any rectangle of 8*4 units and its remaining square–grid–unit of the thematic rectangle right and bottom sides, i.e. 7 of 8 units, and of 4*3 units, represented by thick lines. The 4 units, respectively. The fraction 1/8 was hieratic sign of number 10 that looks like a represented either by a sign that looks like cross section of a pyramid, and that has the nowadays symbol of an angle , which pointed and not arched crest, is not, there - marks one-eights the area of a square of 3*3 fore, imitating the corresponding hiero - units, or 4*4 units. In the decomposition of glyphic arched sign. More likely, it is a 2/37 in the 2/n table in RMP, the fraction 1/8 composite design from two diagonals of 5 was written as two horizontal strokes of units of two attached rectangles, which fol - four units each and a dot of the whole one lows the same principle of representing a above them. number by units but in inclined form. And The thematic rectangular grid of 4*3 units this design idea of the sign of number 10 allows also the generation of various forms might imply that the designer knew well of some integers, e.g., the numbers 4, 6, 7, the right–angled triangle (3, 4, 5), as half the and 9. The hieratic mathematical texts in - rectangular 9 that its dimensions is 4*3 units. clude other forms but all of them are based On the other hand, the fractions 2/3, 1/2, and on the same grid system. For example, in 1/3 were represented, in such a way, to mark MMPp10, number 9 was written to repre - the area deducted from, or remained in, that sent the sum of the parts (5+2+2) instead rectangle. The fraction 1/4 was represented of writing it to represent the sum of the by dividing any square into four quarters, parts (2+2+2+3) ; number 7 was written e.g., a square of 3*3 units. The sign of the in such a way that seems to represent (4 top fraction 1/3, as it some times appears in the +3 bottom) instead of writing it in the nor - hieratic texts, e.g., in the "Egyptian Mathe - mal form as shown in figure-1; and num - matical Leather Roll", as backslash tilted to ber 4 was written as four separate ones 45° could be also the 3 square–units along instead of a horizontal line of four suc - the diagonal of the thematic square of 3*3 ceeding but linked units . In the decom - units. Similarly, the fraction 1/4 were also position of 2/9 in the 2/n table in RMP, written, e.g., in RMPp59, as backslash tilted number 6 was written as three horizontal to 45° which represents the 4 units along the strokes above each other and that each is diagonal of a thematic square of 4*4 units.
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