Review Article - Measurement in Ancient Egypt

(Repoduced from Discussions in Egyptology 30 (1994), 87-100)

Elke Roik, Das Längenmaßsystem im alten Ägypten. Christian-Rosenkreutz-Verlag, Hamburg 1993. 407p, 298 x 210 mm, 106 fig. ISBN 3-929322, DM 129.

It is not often that a piece of research seeks to throw an entirely fresh light on a particular aspect of the ancient Egyptian culture, requiring that a long-established understanding should be cast aside while a new representation - the existence of which has hitherto hardly been suspected - is called upon to take its place. Such is the claim, however, made by the architect Elke Roik, in her study of the linear measuring system in Egypt. Inspired by the finding of an unknown measure of length during the recording of the tomb of Tausret (KV 14), under the direction of Prof. Altenmüller, Dr Roik has developed the theory that this linear measure was in general use in Egypt from the earliest times down to the Roman period. Unlike the royal cubit, it had a length of 65 cm which was divided dyadically into eight parts each of about 8.125 cm, and thus made use of the method of division familiar to the Egyptians through the Horus-eye fractions of the corn measure.

The discovery of this measurement was first reported by Prof. Altenmüller.[1] It seemed to be shown by the intervals between the measuring points marked in red on the walls of the tomb of Tausret, above the axes of the standing figures in the wall decoration. These markings are assumed by Elke Roik to have been connected with the canonical grids, no longer preserved, in which the figures were drawn; and they seem to have been transferred across from other marks which were placed along a red line that was drawn on the ceiling to fix the tomb's central axis. It therefore appeared that the decoration scheme may have been worked out during the construction of the chambers.

Following a description of the cubit system, the corpus of the Egyptian measuring rods and the difficulty of ascertaining a measuring system from the monuments (p.5-28), Dr Roik continues with an explanation of the reasons for believing that a new measuring system has been discovered in Tausret's tomb. She states that the intervals between the red measuring marks fall repeatedly between 30-33 cm, 46-49 cm, and 64-65 cm, and therefore suggest a basic unit of about 16 cm. No statement is made of the actual distances between the individual marks in her description of the tomb (p.30-40); and instead, only theoretical distances are indicated as multiples of the assumed unit of 16 cm, making it difficult to determine whether or how easily these distances might be expressed in terms of the palms and fingers of the royal cubit. It may also be questioned whether one should expect the distances between the figures in a wall-scene to have been simply defined in terms of the royal cubit. Elke Roik acknowledges that when a proportions grid was applied to a wall surface, the grid was scaled according to the space available, so that a continuous variation is found in the measurements of the squares used in different grids. The same principle of scaling could perhaps also have applied to the planned distribution of several figures across a wall surface, thus giving rise to a working module which might well involve some awkward fractions of a cubit.

Nonetheless, since the same module of 16 cm seems to have been used in several chambers, and may also be shown by a series of red marks on the ceiling one of the chambers, spaced 14-19 cm apart, there is some evidence for the existence of a unit which does not fit in easily with the recognised divisions of the cubit. This would appear to be confirmed by the width of the pillars in the burial chamber of Tausret, of about 65 cm; for although 65.5 cm can be expressed in conventional terms as 1 cubit 1 palm 3 fingers, this choice of dimension is not very plausible; and if taken as the basis for the dyadic subdivisions of about 32 cm and 16 cm - as found by repeated halving - then we would have to reckon with fractions a finger. On the other hand, Roik takes no notice of the fact that the royal cubit was obviously used in Tausret's tomb, as for example in the passage-width of 4 cubits, the chamber-height of 5 cubits, and the width of room 'E' of 10 cubits.[2] Now the pillar-width of 65 cm is just 1 1/4 cubits, and belongs to the series 10, 5, 2 1/2, and 1 1/4 cubits, which is found by the repeated halving of the dimension of 10 cubits. Hence the units reported by Roik might be seen as an extension of the cubit system using the dyadic method of division, such as might have been found by the repeated folding and knotting of a measuring cord 10 cubits in length.

Being convinced that the dimension of 65 cm represented a unit of length which could not be reconciled with the royal cubit, however, Dr Roik turns her attention to the possible naming of this unit in the Egyptian texts, and seeks a connection with the non-standard measuring rods of 65-70 cm, which were discovered by Petrie at Kahun and Deshasha, and another similar rod from Lisht. Now it has long been considered that a measurement of between 65 and 77 cm could be equated with a unit known as the nbi, which is seldom mentioned in the Egyptian literature and was generally used to record the amount of work carried out in cutting dykes or excavating tombs. Gardiner conjectured that the nbi might be equal to 1 1/4 or 1 1/3 cubit,[3] and thus gave a name for the measurement of 1 1/4 cubit in the tomb of Tausret.

At this point, however, we have to contend with two recently-published alternative explanations for the nbi measure, both agreeing that the length should be 1 1/3 cubit or 70 cm in opposition to Elke Roik's proposed value of 65 cm, but differing in their interpretation as to the use of this measure. Following his survey of a number of Old and New Kingdom rock tombs, Naguib Victor maintains that the nbi had an architectural significance;[4] while Claire Simon believes that it was connected with the canon of proportion, and was used to determine the size of the grid squares in which the human figure was inserted.[5] The existence of a linear measure of about 1 1/3 cubit is in any case proven for the Middle Kingdom by the three rods from Kahun and Lisht, which vary in length between about 67 cm and 70 cm. These rods are clearly too long to answer to Roik's measure of 65 cm; and since each is divided into seven units, they cannot supply the required dyadic divisions.

The Deshasha rod, on the other hand, was found in a secondary burial of the 18th Dynasty, and would appear to support the Tausret measuring system, if the restoration by Petrie is accepted; for although the length of 66.4 cm is slightly excessive, one half was left plain while the divisions of the other half were grouped by Petrie into four equal parts, each of which was only about 1 mm less than the unit of 8.125 cm which Roik takes to be the 'handbreadth' of her nbi-system. Claire Simon, however, objects that these four parts were obtained by combining eight very irregular divisions,[6] so that the significance of the Deshasha rod for Roik's thesis remains doubtful - the more so since the rod is not preserved and Petrie's description is far from adequate.

Whether any of these non-standard rods can be identified with the nbi measure is another question. Gardiner's suggestion that the nbi might equal 1 1/4 or 1/1/3 cubit was entirely dependent upon his uncritical acceptance of the interpretation put forward by Hayes for the word nbi as it occurs in work records written on ostraca during the cutting out of the tomb of Senenmut.[7] It had previously been established that the nbi was the dynastic Egyptian equivalent of the demotic nbe and Greek naubion, which Thompson had shown to be volume measurements corresponding to a cube with a side of two royal cubits.[8] This unit was employed in demotic ostraca to record the work done in the cutting of dykes; and as shown by Gunn, it had earlier been used in a similar context in an ostracon from Abydos dating to the time of Seti.[9] Whereas Thompson thought that the nbi was originally the stake, two royal cubits in length, which was employed in foundation ceremonies to mark out the ground, and consequently came to be used for the measurement of volumes of earth, Gunn traced the origin to the nbit or reed; and this explanation was accepted by Gardiner.[10]

Hayes did not dispute that the dynastic nbi measure had given risen to the demotic nbe and Greek naubion, but was unable to reconcile the length of two cubits with the value he had derived on the assumption that the nbi measures on one of the Senenmut ostraca were linear dimensions relating to the entrance of Senenmut's tomb. As I have shown in some detail elsewhere,[11] however, the inconsistencies in Hayes' analysis make it certain that he did not understand the procedures followed by the stonecutters in carrying out the excavation of a tomb. First of all, acknowledging that the nbi rod could signify a cubic measurement, Hayes prefixed the word 'cubic' in brackets in five instances in his translation to make this clear. In ostracon n° 62. for example, he rendered lines 5-6 as: '30 men, who did 29 (cubic) rods'.[12] In the four remaining instances of the nbi in these ostraca, on the other hand, Hayes omitted the word 'cubic' because it appeared to him that the measurement was one of three linear dimensions which could be multiplied together to define a volume of rock.

In order to take the nbi as a linear measure in these instances, moreover, Hayes was obliged to vary his translation of the term mdwt, which as shown by Carter and Gardiner in their treatment of the Turin Plan of the Tomb of Ramesses IV,[13] signified a 'depth' measured horizontally into the vertical face of an excavation. This meaning was accepted by Hayes for ostracon n° 75, in which a depth of 3 cubits is stated; but for ostracon n° 62 he had to take the mdwt to mean a vertical depth or height, so as to obtain three dimensions at right angles to each other which could be supposed to represent a volume.

It is also unlikely that the scribes would have employed besides the royal cubit, a linear measure which was greater by just a small fraction, making the calculation of volumes much more difficult than it needed to be. A clue to the correct meaning of the nbi is offered by a difference in the expression of the nbi and cubit measures, which Roik obscures in her brief treatment of these ostraca (p. 43). When the dimensions are given in cubits, the description of the part measured always precedes the numerical data; and yet with the nbi measurements, the order is reversed. This is clearly shown in ostracon n° 75:

It was from this last ostracon that Hayes hoped to place the length of the nbi at between 65 and 77 cm, believing that the 3 nbi in its width', and '7 nbi in its height' could be equated with the dimensions of the doorway of 2.32 m wide and 4.53 m high. The scribes did not, however, state linear dimensions in this manner, but rather as 'width 2 cubits, depth 2 cubits' - as seen again in ostracon n° 76. The difference in the expression of the nbi measures can be explained by the fact that they refer to the volumes of rock removed from the tomb, which the scribe calculated as a record of the work carried out.

When the cross-section of a passage was as large as it is in Senenmut's tomb, the stonecutters would not have taken out the entire cross-section in one operation, as Hayes assumed, but would instead have proceeded in stages. The first step would have been to create a drift-way large enough to stand in, which would have been cut just below the intended level of the roof. This initial passage would then have been widened out to give about the full width, after which the floor would have been cut downwards across the width to give about the full height. The volumes of rock taken out with the enlargement of the passage had therefore to be calculated in two parts, giving the number of nbi removed 'in the width' of the passage, and similarly 'in the height'. A calculation based on the dimensions of the passage and the numerical data on the ostraca leaves no doubt that the nbi was here equal to a cube with a side of 2 cubits,[14] and gave a volume of 8 cubic cubits as acknowledged by Thompson and Gunn. It can also be shown that the initial drift-way measured 2 cubits wide and 4 cubits high; and since one nbi of stone was cut out with each royal cubit of progress into the hill-side, the volumes did not even have to be calculated. When employed as a linear measure, the nbi was simply the double-cubit rod, at least two examples of which still exist.[15]

The rare references to the nbi in the Egyptian literature also make it certain that the linear nbi measure can not have been less than 2 cubits. In 'the hardship's of a soldier's life', we are told that the Egyptian soldier was recruited 'm nhn n nbi', as a child of a nbi.[16] Now Roik translates this phrase as 'Kind von 2 Nbj' and gives a confused and misleading account of it (p.46). After correctly stating that Gardiner originally considered the nbi to be a measure of 2 cubits,[17] indicating a height for the recruits of 1.05 m, she goes on to claim that Gardiner's interpretation would imply a height of 2.10 m or 4 cubits, while a nbi of 0.65 m would give the more realistic height of 1.30 m. But the fact is that a height of only one nbi is stated in the text; and so unless Egyptian children were enlisted when they could barely walk, a length for the nbi of only 65 cm is impossibly small.

The equation between the nbi and the 'non-standard' rods from Kahun, Lisht, and Deshasha, cannot therefore be supported, and the significance of these rods must remain obscure because no name can be assigned to them in the everyday Egyptian language. With regard to Roik's thesis, we have only a rather tenuous connection between the conjectural measuring system in the tomb of Tausret and the Deshasha rod, which will henceforth be referred to as the 'rod-65' to distinguish it from the 'rod-70' of Lisht and Kahun.

It is against this background that we must judge Roik's claim that the rod-65 was the customary Egyptian measurement of length from protodynastic times down to the Roman period. Not content to see the rod-65 as perhaps merely a working measure used by artisans in the decoration of tombs, she contends that it was employed in Egyptian architecture and in artefacts of all kinds. She is determined to throw over the royal cubit as the primary linear measurement in ancient Egypt, in opposition to Lepsius, Petrie, Borchardt, and many others, who have traced the use of the royal cubit in the dimensions of numerous monuments. It is a surprising attack on the system of the royal cubit, the importance of which has seldom before been disputed - the only question for debate having been the precise number of millimetres to be assigned to the length of cubit in use at any given period. Roik maintains, however, that researchers have hitherto assumed that the royal cubit alone was employed in Egyptian architecture, and have therefore failed to notice that an entirely different measuring system was in operation. But although it is true that the significance of the cubit has often been taken for granted, Roik is mistaken in thinking that so many investigators would have persevered with the cubit system if it had not very often yielded satisfactory results.

What, then, of the royal cubit rods discovered in tombs of the New Kingdom and onwards? Roik suggests that these rods were of a special type intended for funeral equipment, as their inscriptions with offering formulae and titles of the deceased clearly show; or else they were votive rods belonging to temple inventories. The conjecture that the length of the royal cubit was sacred and non-functional cannot be dismissed out of hand, and yet there are uninscribed rods of the same length which seem very likely to have been used. In Roik's opinion, these rods were not calibrated with sufficient accuracy to have been functional, and might be ascribed to poorer burials.

Dr Roik also points out that the designation 'royal' for the cubit is first known from the New Kingdom, and questions whether earlier references to the cubit without this designation should be assumed to indicate a length of seven palms, as opposed to the six palms of the small cubit. She cannot deny that records of linear measurements in Egypt invariably involve the cubit system; and as cubit and nbi measures are mentioned side by side on the Senenmut ostraca, she infers that the cubit and the nbi must have been related. As we have seen, the linear nbi was in reality just two cubits; but despite the fact that the measure of 65 cm can be placed in the simple proportion of 5:4 to the royal cubit, Dr Roik describes this as a scheinbare or apparent connection, because the relationship is not commensurate with the dyadic divisions of the rod-65. Instead, therefore, she derives a 'cubit-48' containing six of the eight units of the rod-65, and finds a length for this conjectural cubit of 6/8 x 65 or 48.75 cm. This length and division cannot, of course, be reconciled with the seven-part cubit rods provided as burial equipment, nor is it compatible with the dimensions in the Turin Plan, or with the seked problems in the Rhind mathematical papyrus, which all involve a cubit of seven palms without being specified as 'royal', as Dr Roik acknowledges. In addition, the measurements in the Reisner papyri also require a seven-part cubit, and confirm that this measure was used without the designation 'royal' during the Middle Kingdom.

Again, regardless of how many palms the functional cubit is supposed to have contained, the exact length had been determined even before the first cubit rods had been discovered. Elke Roik points out that Sir Isaac Newton had derived the cubit from the measurements taken inside the Great Pyramid by John Greaves in 1646, but wrongly asserts that he obtained two different cubits from the pyramid (p.18). In fact he found only one cubit, and showed that the Kings Chamber measured 10 by 20 royal cubits, that the passages were two cubits wide, that the Grand Gallery was four cubits wide, and that the ramps in the Gallery were one cubit wide and high. Since, furthermore, the Queen's Chamber measures just 10 by 11 cubits, the niche in the east wail is just two cubits deep, and the width of the niche steps inwards from three cubits at the base to one cubit at the top, the general use of the cubit in the passages and chambers can hardly be doubted; and Newton was of the opinion that 'the Pyramid was built throughout after the measure of this Cubit.'[18]

It seems curious, therefore, that Dr Roik should take the dimensions of the Kings Chamber as a documentary example for the use of the rod-65. This is made possible because the proportion of 5:4 with the royal cubit means that the multiples of five cubits in the width and length of the chamber are also multiples of four rods-65. It also so happens that the height of the chamber is not a whole number of cubits, and is close to nine rods-65. Yet as Petrie discovered, the height is explained by being exactly equal to half the diagonal of the floor, so that the end-wall diagonals of the chamber measure 15 cubits, while the cubic diagonal is 25 cubits; but again, these multiples of 5 cubits can be expressed as whole numbers of the rod-65. The use of the royal cubit is shown exclusively, however, in the dimension of 2 cubits, which occurs very frequently as the passage-width in this and other pyramids, and as the usual pillar-width both in the temples of Giza and in the royal tombs at Thebes. For this dimension Roik can only offer the number of 13 units of the rod-65, or 13 x 8.125 cm - a value so implausible in view of the simplicity obtained with the royal cubit, that it should be obvious which measurement was used. The dimension of 2 cubits - the true nbi-measure - was itself often doubled to give, for example, the shaft-width of 4 cubits in the mastabas at Giza.

By dismissing the royal cubit as if its employment in Egyptian monuments was a matter of interpretation, however, Roik makes a serious omission. In the course of their preparations, the builders of the Old Kingdom often marked up wall surfaces with levelling lines at cubit intervals, giving next to these lines the heights in cubits above or below a fixed datum. At Meydum for example, Alan Rowe discovered horizontal lines on the face of an inner casing marked as "six cubits" and "eight cubits", and observed that the cubit in use was 52.5 cm.[19] Similar lines were found by Petrie in the foundations of the nearby mastaba no 17, using the same cubit and giving the levels in cubits below the zero-datum.[20] At Giza, Petrie noted cubit markings in the relieving chambers in the Great Pyramid;[21] while Reisner recorded series of levelling lines on the walls of the Menkaure pyramid-temple, marked with numbers of cubits in the same manner.[22] At Abusir, Borchardt found markings of exactiy the same type on the core-masonry of the Pyramid of Neuserre.[23]

Given the fundamental nature of these evidences for the use of the royal cubit during the Old Kingdom, it is unfortunate that Dr Roik ignores all of them. She does, however, refer to the comparable evidence for the cubit of Middle Kingdom, which is shown in the biographical inscription of Khnumhotep from tomb n° 3 at Beni Hassan (p.370); but she is convinced that the units of the rod-65 were employed in this tomb as elsewhere. While at first dismissing the dimensions in the Turin Plan of the Tomb of Ramesses IV as a reckoning up of the decoration (p.173), on the other hand, she seems finally to concede that the seven-part royal cubit must have been used (p.375), as indeed the correspondence between the dimensions specified in the plan and those used in the tomb itself, allows no other conclusion.

Roik's conviction that the rod-65 was employed in Egyptian architecture is partly to be explained by her chosen method of analysis. We have already seen that besides the 'rod-65', she has conjectured a 'cubit-48' (her own nomenclature), which is composed of six units of 8.125 cm roughly comparable to the palms of the royal cubit. In addition, using Petrie's analogy between the non-standard rods and the double-foot of Asia Minor, she invokes a foot measure containing four units of the rod-65. She is vague about which of these lengths was used as the primary measurement, and neatly side-steps the issue by basing her analysis on the unit of 8.125 cm, noting when the number of units in any given dimension happens to coincide with a whole number of feet, cubits-48 or rods-65. Her dimensional material is presented in a large number tables, in which are extracted the metric measurements of monuments of all kinds - tombs, temples, stelae and sarcophagi - with the nearest equivalent number of units, the corresponding metric value, and the number of rods, cubits or feet if appropriate (p. 232-369). No explicit statement is made of the errors involved, and it is left to the reader to decide whether the metric equivalent for the assumed number of units, corresponds with sufficient accuracy to the measured dimension. Nor has the attempt been made to derive a mean value for the unit considered to have been used in any monument.

Elke Roik defends this approach by arguing that the ascertainment of a measuring system must be based upon small distances, in which the error of execution as well as the modern survey error may he supposed to be small. She therefore attempts to confine her attention to parts of buildings such as doorways, pillars, and the spaces in between, and expresses the dimensions in terms of the small unit of 8.125 cm. There is a flaw in this argument, however, because the basic requirement for a dimensional analysis is not that the dimensions should be small, but that the probable errors in the dimensions should be much smaller than the unit which is to be detected. Other factors being equal, it is no easier to discern the number of palms in a distance of 5 metres, than the number of cubits in a distance of 35 metres. With small distances, however, residual errors arising from the variable or incomplete dressing of stone surfaces, the shifting or weathering of masonry, and the approximations of modern measurements, will represent a much larger fraction of a unit of only 8 cm, than of a unit of say 50 cm; and it is therefore more sensible to base an analysis on a length such as the cubit, whether the royal cubit or 'cubit-48', than on the small 'handbreadth' of the rod -65.

If short distances are to be analysed in terms of a small basic unit, then the original workmanship and the available survey data must be sufficiently precise. To make this clear, we can consider the result when a unit of 8 cm is applied to a set of random measurements. No measurement will differ by more than 4 cm from the nearest whole number of units, some measurements will by chance correspond to a whole number almost exactly, and the average error obtained with such random data will be 2 cm or 25%. To show that the use of a unit of 8 cm in a given set of measurements is significant, therefore, the 'mean quantization error' must be much less than 2 cm; and it may well be less than 10% if a genuine correlation exists. Almost invariably, however, when the sets of data provided by Roik are analysed, the quantization errors for the unit of 8.125 cm average about 20-30%, thus showing that a significant correlation does not exist.

Dr Roik's unscientific treatment of the data is further illustrated by the one instance where she does attempt a statistical analysis, for the courses of the Great Pyramid. Equating the lowest 50 course-heights with the nearest whole number of units in each case (p.272), she finds that about 50% of these courses show a deviation of 2 cm. But although this mean error seems to her small enough to show a correlation, it is in reality merely the expected result when the unit of 8.125 cm is applied to a randomly distributed set of data. The formal methods which can be used to detect quanta in sets of dimensions in fact show that the lower courses of the Great Pyramid were so carefully graded into successive diminishing sequences, that no quantum can be found; yet towards the summit, these sequences give way to a more uniform system in which about thirty courses are each one royal cubit in thickness - a fact which is glossed over by Roik.

It must also be noted that Dr Roik has made no attempt to show that the units of the rod-65 are more significant in the measurements she gives, than the palms of the royal cubit. She does not, therefore, present an argument as such, since a comparative analysis is not offered; and except in a few cases, we are left with a dogmatic statement of the numbers of units resulting from the mechanical conversion of the survey data. Scant effort has been made to explain the choice of dimensions, which appear rather as arbitrary values in which the use of the wrong unit has obscured the relationships which often exist between the components of a structure. It is, however, with reference to the proportions that the fractions of a cubit which were sometimes chosen may be discerned, when the accuracy of the data is not otherwise sufficient to resolve the question. Such is Roik's determination to promote the units of the rods-65, however, that she finds it possible to list the dimensions of a sarcophagus as 6 1/2, 13, 26 and 39 units (A2, p.317), without giving notice of the fact that these dimensions are exactly 1, 2, 4 and 6 royal cubits.

In order to determine whether the unit of 8.125 cm was statistically more significant than the palm of 7.5 cm, some hundreds of the dimensions provided by Roik were analysed by computer. The 'mean quantization errors' resulting from this investigation showed that in most instances, neither the palm nor the unit could be proven to have been used, for the probable reason that the data are mostly not sufficiently accurate for a small quantum to be detected. The analysis was designed, however, to reveal any quanta in a given range, and yielded some quite unexpected results. In particular, the dimensions of a sarcophagus of the Third Dynasty were found to be so strongly quantized in terms of a unit of 8.72 cm, that there appeared to be no doubt that this unit had been employed in the manufacture. For the eight dimensions listed by Roik for this sarcophagus ((Al, p.317), the mean quantization error was only 7.6%, with a mean execution error of only 1% in the dimensions overall.

This unit of 8.72 cm at first appeared to be an artisans palm, measured across the full width of the palm as opposed to the width measured across the fingers; but it is also just one-sixth of the royal cubit, and thus suggests that the division of the cubit which occurs on cubit rods of the Graeco-Roman Period, and which is known as the 'great palm', was in use as early as the Third Dynasty. A similar conclusion had in fact been reached by Junker, as a result of his measurements in the Giza mastaba-tombs; for although Junker knew that dimensions in Egyptian texts used a seven-part cubit, he detected a palm of 1/6 cubit or 8.75 cm;[24] and this finding is certainly justified by the data. In addition, the twenty-two measurements provided by Naguib Victor as proof of the use of the 'rod-70' in a tomb of the Sixth Dynasty, in fact offer compelling evidence that the royal cubit was here divided into thirds. None of Victor's data involve the seven-part division of the rod-70, and yet many of them require a division into halves or quarters, thus giving fractions which are in fact 2/3 and 1/3 of the royal cubit.[25] All of the dimensions listed by Victor can be explained very accurately using these fractions of the cubit.

At the same time, numerous small measurements in the Pyramids of Giza bear testimony to the use of the seven-part royal cubit, and it now appears certain that these two systems of division must have existed side by side. The simplicity of the six-part system, which contained the important natural fractions 1/2 and 1/3, suggests that it preceded the seven-part division, which was based on the width of four fingers and was probably a refinement introduced for religious reasons. Precisely why the seven-part cubit was employed in particular monuments is otherwise unclear, since it does not seem to have been specifically royal. A study of the dimensions of the temple of the Pyramid of Meydum, for example, and in the White Chapel of Senwosret I at Karnak, has again shown the use of the six-part royal cubit. That quarters, halves and thirds of this cubit must have been used alongside the seven-part division is in any case proven by the measurements in the Reisner papyri;[26] and it is clear that whilst the length of the royal cubit was maintained, the division of this length was in practice flexible. The assumption of Elke Roik, that the use of the royal cubit must always have involved a palm of 1/7 cubit or 7.5 cm, is therefore not valid.

Canon and Metrology
Having cast aside the royal cubit and its divisions as the primary system of measurement in Egypt in favour of the rod-65, Dr Roik is obliged to consider the implications for the Egyptian artists canon of proportions, since together with Erik Iversen and many others, she maintains that the grids in which the canonical human figures were inserted, were closely connected with metrology. She discusses this question at some length (p.88-l67), and provides a useful summary of the various theories which have been put forward, illustrated with line drawings reproduced from some of the many contributions to the subject which have been published since the time of Lepsius.

It is Roik's contention that the size of the grid square which was used to obtain the correct proportions, was in full-size representations always equal to the unit of 8.125 cm. She equates this unit with the handbreadth, but fails to account for the fact that in the early canon - which she calls the canon-18 after the 18 squares placed between the soles of the feet and the hairline in standing figures - the width of the square appears to include the thumb with the palm, and should accordingly have been larger than the dimension she has assigned to it. She believes, however, that the variations in the canon which took place during the New Kingdom - including a 'canon-l9' for the Amarna Period, and a 'canon-20' for some subsequent reigns - were the result of a continuous process which led eventually to the introduction of the 'canon-21' at the beginning of the Saite Period. In her view, the size of the grid square cannot have been altered with each of these canonical changes; and hence, assuming that a rigid connection existed between the squares of the grid and anatomical measurements, she concludes that these variations were the result of a progressive increase in the height of the Egyptian people.

If there is any evidence for this growth, however, it is not provided by Roik; and she makes no attempt to demonstrate such an increase in stature from the anatomical measurements of mummies, which she lists in chronological order (table 21). Measuring to the crown of the head, the average height of the ancient Egyptian male should have increased from about 19 × 8.125 cm or 1.54 m for the canon-18, to perhaps 22.5 × 8.125 cm or 1.83 m for the late canon - a range which is impossible to reconcile with the anatomical data. Roik assumes that the Egyptian artists interpreted the size of the grid square as an absolute or 'full-size' dimension, and that the canon was repeatedly recreated to allow for anatomical changes; and yet the essential function of the grid was certainly to establish, not the absolute dimensions of the human figure, but a set of proportions which would remain acceptable regardless of the stature of the individual being portrayed. The Egyptians therefore had no reason to revise the canon to allow for variations in height, even assuming that the grid square had initially been connected with an absolute dimension. Since the scale of reproduction was continuously varied, and canonical figures were probably handed down on papyri, writing-boards, or ostraca, in which the grid square was some arbitrary small fraction of its original size, it is by no means certain that the later Egyptian artists had a clear conception of the exact full-size dimension of the grid square.

The changes in the canon, especially during the Amarna Period, were of course stylistic variations intended to achieve particular effects, and would almost certainly have been devised in preliminary sketches without regard for anatomical measurements. The occasional placing of subordinate figures in the same grid as the main figure suggests that the grid was sometimes used as the copying device for a composition which had been worked out in a sketch; so that although the grid would have helped the artist to obtain the desired proportions in the sketch, it also enabled the draughtsman to reproduce faithfully in the wall-scene, any free-hand or subsidiary elements which the artist had introduced. The traces of grids in some scenes, therefore, need not always mean that a canon was being imposed, nor should we overlook the ability of the Egyptian artist to draw consistently-proportioned figures without the use of a grid.[27] Following the Amarna Period, in any case, the orthodox canon-l8 continued to be widely employed.

Although for these reasons Elke Roik's thesis cannot be accepted, she at least draws attention to the flaws in Iversen's hypothetical metrology, and points out that although Gay Robins has plausibly determined the size of the grid square from anatomical measurements, the resulting dimension of 1 1/5 palms of the royal cubit is no more convincing as a metrological unit than Iversen's fist of 5 1/3 fingers. Robins has asserted that the palm of four 'metrological' fingers was equivalent to five 'natural' fingers, and that the thumb was added as a sixth finger to give a fist of 6/5 palms;[28] but in this case, taking the palm of 7.5 cm, the width across four natural fingers would presumably have been 7.5 x 4/5 or only 6 cm - a result which is far too small. It is clear, on the contrary, that the so-called palm or shesep was measured across the fingers, so that the metrological and natural fingers were about the same. Robins has recently connected the placing of five grid squares in the length of the forearm from the elbow to the fingertips, with the ratio of 2:3 by which this length is approximately divided at the wrist;[29] but it seems uncertain whether she regards this as an explanation for the size of the grid square of 1 1/5 palms, when the length of the small cubit of six palms might otherwise have placed six squares of the grid in the forearm.

By equating the grid squares with the units of the rod-65, Elke Roik has attempted a metrological simplification which has been noticeably lacking in earlier canonical theories; and yet it must be conceded that the Egyptian canon of proportions need not necessarily have had a metrological foundation. To institute the system of horizontal guidelines which was employed during the Old Kingdom, it would have been sufficient to stand someone up against a wall with their arms hanging down, and to mark off the levels of the significant points on the wall surface at the knees, wrists/buttocks, elbows, shoulders and hairline. By stretching a cord between the different points, through a process of trial and error, it would have been found that the height from the floor to the hairline could be divided into three equal parts at the knees and the elbows. The simple process of halving by folding the ends of the length of cord together would then have shown that the point for the wrists/buttocks was at 3/4 of the elbow level, which in turn was at 3/4 of the shoulder level. Expressing these different levels in a single series of fractions, the wrists/buttocks could be placed at 3/4 × 3/4 or 9/16 of the shoulder level, the elbows at 3/4 or 12/16, the knees at 6/16, and the hairline at 18/16. Consequently, the fundamental Egyptian processes of cord-stretching and halving would have divided the human figure into exactly those fractions which are given by the 18 squares of the grid, and the size of the square would have arisen naturally from a comparison of the different intervals. The distances between the elbows, wrists, and shoulders thus gave a unit equal to 4/16 - 3/16 = 1/16 of the shoulder level, and so 1/18 of the hairline level (fig. 1). The canon of proportions could thus have been created without any reference to absolute measurements, and the premiss of Roik and Iversen that the canon was based on metrology, is shown to be unwarranted.

It is, on the contrary, conceivable that Egyptian metrology derived from the canon, given that the royal cubit is much longer than the length of the forearm to the fingertips and is not the anatomical unit Iversen supposed it to be. From the measurements of some 60 mummies, Gay Robins has shown that the average height of the adult Egyptian male was 166 cm,[30] so that the mean height to the hairline must have been about 18/19 × 166 or 157.3 cm. Each of the three equal parts into which this distance was divided at the knees and elbows, therefore, was equal to 52.4 cm, which is exactly the length of the royal cubit. Each part, furthermore, contains six units of the grid, and thus conforms to the six-part division of the royal cubit which we have now traced back to the Old Kingdom. The royal cubit may thus be explained as the true 'canonical' cubit, which may have been introduced because although the small cubit represented the natural length of the forearm, it divided awkwardly about 3 1/2 times into the height up to the hairline. Being based upon six natural palms of 8.74 cm, the royal cubit divided exactly three times into the canonical height, and each of the 18 grid squares in the height could be equated with the palm. If the grid square sometimes takes in the thumb, it is because the width of the hand was drawn slightly too small in relation to the rest of the body - an error of proportion which is even more obvious in the oversized feet of Egyptian figures. In many instances, however, the grid square is clearly narrower than the 'fist', and closely corresponds to the full width of the palm.[31]

The units of the six-part royal cubit were not, therefore, the great palms of a reformed cubit as Iversen supposed, but the natural palms of a division of the cubit which had existed at least since the Old Kingdom. Of these palms, five are contained in the small cubit, and correspond to the five grid squares in a length of forearm of 5/6 × 52.4 cm or 43.7 cm. This supports Petrie's observation that the small cubit as seen on some cubit rods was shorter than the length of six shesep, or 45.0 cm, being at most 23 fingers and not 24 fingers as is so often stated.[32] The six divisions from the seven-part royal cubit were perhaps merely the nearest equivalent to the small cubit in the more refined seven-part system, which was based not upon palms or handbreadths, but on the shesep or width across four natural fingers. It also now follows that the canonical reform of the Saite Period cannot have been caused by a reform of the cubit, which is in any case unlikely to have been introduced by Egyptians who endeavoured to return to the prototypes of the Old Kingdom. The reform can, however, be explained as a false archaism, since in attempting to re-establish the canon after the alterations of the New Kingdom, it was probably assumed that the 'sacred' seven-part division of the cubit should be used, so that seven squares each equivalent to the shesep now took the place of every six squares in the original grid.[33] The canonical height was now about 22.5 × 7.5 equals 169 cm, or about 3 cm greater than the original height; and the level of 21 squares or three royal cubits was placed slightly below the hairline.

We can, finally, endorse Roik's assertion that a six-part cubit was in general use in Egypt, not as the small cubit or as six-eighths of the rod-65, but as a system of division of the royal cubit with a length of about 52.4 cm. It is also clear that Roik is justified in her belief that the seven-part cubit rods from tombs give an incomplete view of the customary usage. The 'sacred' seven-part system was perhaps originally intended for the building of temples, and was naturally selected for burial equipment, although already during the Middle Kingdom it was being used for mundane purposes alongside the more convenient divisions of the cubit into halves, thirds and quarters. In the Cairo Museum, according to Roik (p.24), there are three functional six-part wooden rods each 52 cm in length, which might belong to this customary cubit system. It seems possible, however, that they were classified by Petrie as Assyrian or Jewish, so that their significance for the Egyptian measuring system has not hitherto been appreciated.

If Dr Roik has a claim against the system of the royal cubit, it is because the flexibility of this system has been so poorly understood. The fact that it has been thought possible to challenge the cubit system with the hypothesis of the rod-65, which has no more real scientific evidence to support it than Iversen's metrology or the pyramid-inch, can only be ascribed to the regrettable lack of any formal study of the use of the royal cubit in Egyptian monuments in general; for although the analysis of the dimensions of tombs and temples should have proceeded as a matter of course, and might by now have resulted in a clear understanding of the principles of design employed in Egyptian architecture, the subject has been neglected far too often.

John A.R. Legon

1. H. Altenmüller, SAK 10 (1983), 3-4.
2. Altenmüller, op.cit. 4, gives passage dimensions of 210-214 by 258-264 cm.
3. A.H. Gardiner, Egyptian Grammar (Oxford, 1957), 199.
4. N. Victor, GM 121 (1991), 101-110.
5. C. Simon, JEA 79 (1993), 157-177.
6. Simon, op.cit. 169; W.M.F. Petrie, Deshasheh (London, 1898), 37.
7. W.C. Hayes, Ostraka and Name Stones from the Tomb of Sen-mut (no. 71) at Thebes (New York, 1942), 36-7.
8. A.H. Gardiner, H. Thompson and J.G. Milne, Theban Ostraca (London, 1913), 26, n. 3.
9. In H. Frankfort, The Cenotaph of Seti I at Abydos (London, 1933), 92-4. 10. A.H. Gardiner, Hieratic Papyri in the British Museum, Third Series (London, 1935), I, 42, n.8.; Ancient Egyptian Onomastica (Oxford, 1947), 1, 67, n.1.
11. J.A.R. Legon, GM 142 (forthcoming).
12. Hayes, op.cit., 21.
13. H. Carter and A.H. Gardiner, JEA 4 (1917), 130-158, 138.
14. See note 11.
15. Simon, op.cit. 161.
16. A.H. Gardiner, Late Egyptian Miscellanies (Brussels, 1937), 26.
17. See note 10.
I8. In C.P. Smyth, Life and Work at the Great Pyramid, Vol II. (Edinburgh, 1867), 340-66, 348.
19. A. Rowe, Museum Journal, Vol. XXII no.1 (1931), 23, Pl X.
20. W.M.F. Petrie, Medum (London, 1892), 12-13, P1. VIII.
21. W.M.F. Petrie, The Pyramids and Temples of Gizeh (London,1883), 93-4.
22. G.A. Reisner, Mycerinus, The Temples of the Third Pyramid at Giza (Cambridge, Mass., 1931), 77-8, Pt. XI, XII.
23. L. Borchardt, Das Grabdenkmal des Königs Ne-user-Re (Leipzig, 1907), 154-5, Abb. 129.
24. H. Junker, Giza I (Vienna, 1929), 85-6.
25. N. Victor, op.cit. 105.
26. W.K. Simpson, Papyrus Reisner I (Boston, 1963), sections G,H,I.
27. E. Mackay, JEA 4 (1917), 80.
28. G. Robins, GM 59 (1982), 61-75, 65.
29. G. Robins, JEA 80 (forthcoming).
30. G. Robins, op.cit. (n.28), 70.
31. For example, see G. Robins, Proportion and Style in Ancient Egypt Art (London, 1994), 51-3; figs. 2.7, 2.11, 2.15, 2.16.
32. W.M.F. Petrie, Ancient Weights and Measures (London, 1926), 41.
33. R. Hanke, ZAS 84 (1959), 113-9; Abb. I.