Chap. 20. Value of the Cubit and Digit Pages 178 - 181
Sec 136. The cubit in the Great Pyramid 178
Sec 137. The cubit in other buildings 179
Sec 138. Divisions of lists in the tombs 179
Sec 139. Decimal division of cubit 180
Sec 140. Values of the digit 180
Sec 141. Comparison with previous results 181
136 . p 178. The measurements which have been detailed in the foregoing pages supply materials for an accurate etermination of the Egyptian cubit.
From such a mass of exact measures, not only may the earliest value of the cubit be ascertained, but also the extent of its variations as employed by different architects.*
* On the facade of one of the tombs at Beni Hassan there is a scratch left by the workman at every cubit length. The cubit there is a long variety, of 20.7 to 20.8.
There is no need to repeat here all the details of each case already given, nor to enter on the principles of the determination of units of measure from ancient remains, which I have fully described in "Inductive Metrology."
For the value of the usual cubit, undoubtedly the most important source is the King's Chamber in the Great Pyramid ; that is the most accurately wrought, the best preserved, and the most exactly measured, of all the data that are known. The cubit in the Great Pyramid varies thus :
By the base of King's Chamber, corrected for opening of joints
By the Queen's Chamber, if dimensions squared are in square cubits
By the subterranean chamber
By the antechamber
By the ascending and Queen's Chamber passage lengths (section 149)
By the base length of the Pyramid, if 440 cubits (section '43)
By the entrance passage width
By the gallery width
The passage widths are so short and variable that little value can be placed on them, especially as they depend on the builder's and not on the mason's work. The lengths of the passages are very accurate data, but being only single measures, are of less importance than are chambers, in which a length is often repeated in the working. The chamber dimensions are rather variable, particularly in the subterranean and Antechamber, and none of the above data are equal in quality to the King's Chamber dimensions. If a strictly weighted p 179 mean be taken it yields 20.620 .004; but taking the King's Chamber alone, as being the best datum by far, it nevertheless contracts upwards, so that it is hardly justifiable to adopt a larger result than 20.620 .005.
137 . In the Second Pyramid the base is very indirectly connected with the cubit, so that it is not probable that reference was made to the cubit, but only to the King's Chamber or passage height which are derived from it.
The cubit found varies thus :
By the tenth course level
By the first course height
By the passage widths
By the great chamber, dimensions squared being in square cubits
By the lower chamber, sides in even numbers of cubits
The course heights and passage widths are less likely to be accurate than the chamber dimensions; a strictly weighted mean is 20.68 .03; but, considering the details, probably 20.64 .03 would be the truest determination of the cubit here.
In the Third Pyramid. the work is very rough in comparison with the preceding, and the cubit is correspondingly variable.
By the base
By the course heights of granite
By the first chamber
By the second chamber
By the granite chamber
Here it is evident that the courses are all too thin on an average, though varying from 36.0 up to 42.8, and they are certainly not worth including in a mean. The average of the other elements, duly weighted, is 20.76 .02, or the simple average (as the previous gives scarcely any weight to the chambers) is 20.7I .02, which may be most suitably adopted.
Arranging the examples chronologically, the cubit used was as follows :
Great Pyramid at Gizeh,
Second pyramid at Gizeh
Granite temple at Gizeh
Third Pyramid at Gizeh
Third Pyramid peribolus walls
Great Pyramid of Dahshur
Pyramid at Sakkara
Fourth to sixth dynasty, mean of all
Average variation in standard
138 . Besides these examples of the cubit as used in various buildings, the other principal unit of early times---the digit---may be obtained from the tombs. A usual feature of the decoration, of both the rock-cut tombs and the built p 180 Mastabas, is a list of offerings ; these are usually written in a tabular form, in a number of square spaces like a chessboard. It was therefore necessary for the workman to set out a series of equal spaces, and to do this he most naturally marked off an even number of whatever small measure he was in the habit of using; just as an English workman would mark off a number of spaces of three or four inches each, rather than adopt any irregular fractions. These lists are particularly valuable in giving the unit, for two reasons first, the designer was usually unfettered by limits, he could make the spaces a little larger or smaller, so as to work in round numbers, and he could mark off the lengths on a smooth wall with great nicety; and, secondly, the repetition of so many equal spaces gives an admirable means of ascertaining where the workman's errors lay, and also gives such a number of examples that the exact quantity may be very accurately determined.
In the examples of such lists measured by me, the regularity of the spaces deserves notice; the errors of workmanship, shown by the average variations of the lengths from the mean of the whole, stand thus :
Tomb no. 15 of Lepsius, Apa.
Tomb no. 68
Tomb W. of Great Pyramid, Ka-nofer-a
Tomb No. 64
Tomb under 69
Tomb, (Painter's canon) Ases-kaf-ankh
Tomb, (Painter's canon) Ases-kaf-ankh
Here it is seen that, with one exception, the error of marking the spaces one over the other is greater than that of marking them side by side. Omitting the tomb of Ka-nofer-a (which was never finished, and is merely in the first blocking out), the mean error in width is .028, and in height .040. Considering that the engraved lines are usually about 1/10 inch wide, a mean error of 1/25 or 1/35 inch on the length of the spaces shows very careful work.
139 . Of these engraved lists the first two have a unit of a decimal division of the cubit; in No. I the spaces are 16/100 of a cubit wide, and 20/100 high, or 4/25 and 5/25; and in No.2 the spaces are 14/100 wide, and 16/100 high, or 7/50 and 8/50. The cubit of No. I would be 20.45 .05, and of No.2, 20.58 .08. This is of course inferior as a cubit standard to the determinations from large buildings but it is very valuable as showing the decimal division of this cubit, which is also found in other countries.
140 . Of the digit there are three good examples : No.3, squares of 8 digits wide, 9 digits high, mean digit .739 .004
No.4, squares of 4 digits wide, 5 digits high, mean digit .728 .001
No.5, squares of 3 digits wide, 4 digits high, mean digit .722 .002
Weighted means .727 .002
p 181. The squares of No.6 may be designed in digits, but they seem to be irregular, being 3.600 .004 wide, and 3.775 .012 high. In the same tomb of Ases-kaf-ankh, are two subjects with the artist's canons, drawn over the finished and painted subject. What the purpose of this can have been it is not easy to say but the squares in one subject are .4675 .015; and in the other 1.407 .005, or three times .4690; the mean of both is .468 .001, which is not simply connected with any other quantity of digits or cubits.
141. The values of the cubit and digit, found in use in the cases mentioned in this chapter, agree remarkably closely with what has been already worked out. For the cubit I had deduced (Inductive Metrology, p.50) from a quantity' of material, good, bad, and indifferent, 20.64 .02 as the best result that I could get; about a dozen of the actual cubit rods that are known yield 20.65 .01; and now from the earliest monuments we find that the cubit first used is 20.62, and the mean value from the seven buildings named is 20.63 .02. Here, then, by the earliest monument that is known to give the cubit, by the mean of the cubits in seven early monuments, by the mean of 28 examples of various dates and qualities, and by the mean of a dozen cubit rods, the result is always within 1/50 inch of 20.63. On the whole we may take 20.62 .0I as the original value, and reckon that it slightly increased on an average by repeated copyings in course of time.
The digit, from about a dozen examples deduced from monuments (Ind Met., p.53), I had concluded to be .7276 .001 ; here, from three clear and certain examples of it, the conclusion is .727 .002 for its length in the fourth dynasty, practically identical with the mean value before found.
As I have already pointed out (Ind. Met., p. 56), the cubit and digit have no integral relation one to the other ; the connection of 28 digits with the cubit being certainly inexact, and merely adopted to avoid fractions. Now these earliest values of the cubit and digit entirely bear out this view ; 28 of these digits of .727 is but 20.36 .06, in place of the actual cubit 20.62 .01. Is there then any simple connection between the digit and cubit? Considering how in the Great Pyramid, the earliest monument in which the cubit is yet found, so much of the design appears to be based on a relation of the squares of linear quantities to one another, or on diagonals of squares, it will not be impossible to entertain the theory of the cubit and digit being reciprocally connected by diagonals. A square cubit has a diagonal of 40 digits, or 20 digits squared has a diagonal of one cubit; thus a square cubit is the double of a square of 20 digits, so that halves of areas can be readily stated. This relation is true to well within the small uncertainties of our knowledge of the standards; the diagonal of a square cubit of 20.62 being 40 digits of .729, and the actual mean digit being .727 .002. This is certainly the only simple connection that can be traced between the cubit and digit; and if this be rejected, we must fall back on the supposition of two independent and incommensurable units.