Published in Annales du Service des Antiquites de L'Egypte,
Volume 26, 1926, pages 197 - 202
Printed by the Institute of France, Oriental Archaeology,
Le Caire, France
Plate appended to volume showing the Ostracon and side view of vault.
"In April 1925, the excavations of the Department of Antiquities at the Step Pyramid brought to light the object reproduced by a photograph in the plate, and by a tracing in fig 1. It is a limestone flake, apparently complete, 15 x 17.5 x 5 cm., inscribed on one face in red ink. The surface is rough, and the writer's pen has jumped over many small depressions, giving the lines and signs a broken appearance. In some places, also, small portions of the surface seem to have scaled away. There appears to be nothing in the paleographical features of the inscription, or in the contents of the latter, to conflict with the dating which is most probable on general grounds, namely to the period of Zoser (2650 BC).
A transcription of the ostrakon is given in fig. 2. It will be seen that in the five spaces formed by the curve and the vertical lines are lineal measures expressed in cubits, palms and fingers *, as follows :
* One cubit = 7 palms = 28 fingers; one palm = 4 fingers.
It seems clear that each measurement refers to the height of the vertical line to the left of it, and that these lines must be taken as all rising from the same level; the horizontal datum line which according to our ideas would be necessary at the bottom of the diagram is however absent, being doubtless taken for granted. The vertical lines will thus be offsets (in effect a series of coordinates) which by their lengths determine the positions of a series of points on the curved line. Obviously, for this diagram to be of use, the distances of the vertical lines from one another must be known; these data are however not given. It is natural to suppose that the vertical lines are intended to be equidistant, in spite of the inequalities of the diagram in this respect; and the very fact the distance is not specified makes it most probable that it is to be understood as one cubit, an implied unit found elsewhere. (Cf. my remarks Journal of Egyptian Archaeology, XII, 134.) This granted, it is then a reasonable supposition that the curve drops to the zero level at a point one cubit to the right of the shortest vertical line.
Fig. 3 is a diagram embodying these assumptions; the vertical lines are proportional to the lengths stated on the ostrakon (all measurements are given in the convenient unit of the "finger"), and a curve has been drawn to touch the upper ends of these lines and to drop to the zero level on the right. It will be seen that the curve thus obtained gives a fairly good form, and is very similar to that of the ostrakon, except that it drops less sharply than does the latter between "41" and "0"; this difference however probably arises merely from the roughness of the drawing on the ostrakon. The curve does not appear to be part of an ellipse.
(Refer to my proposed mathematical solution in the accompanying paper, An Ancient Egyptian Catenary Construction Curve, where I explain the difference in "sharpness" of the drop. EPM)
The question now arises what may have been the significance of this diagram. The spot at which it was found offers an answer to this. On the west side of the more northerly of the two tombs with facades which are situated by the north-east corner of the Pyramid (see Annales du Service, XXIV, 122 foll.), is an elevated horizontal surface on which the remains of a solid saddle-back construction; a photograph of this on the southern side, which is the better preserved, is given in the plate. Close to the south-east corner of this construction, on its original ground-level, the ostrakon was found. As soon as the latter was transcribed Firth pointed out that it was probably a working diagram for the building of the saddle-back to a desired curve.
This view seems prima facie extremely plausible. For one thing, the
plotting out of a large architectural curve, other than circular, by means of
offsets is just the method that we should expect Egyptian builders to employ;
for another, it is an obvious economy to give the offsets of only one half of a
symmetrical curve. A minor point is also that the diagram is written in red,
which is the colour used for almost all early construction memoranda and
diagrams found on masonry. To test the accuracy of this theory as fully as
possible, it is necessary firstly to draw, as is done in fig. 4, a symmetrical
figure of which the curve established in fig.3 is equal to the right-hand half,
and then to ascertain how far that figure, and the dimensions stated on, or
inferred from, the ostrakon, agree with the actual saddle-back.
It will be seen that fig.4 gives a curve which is 10 cubits wide by 3 1/2 cubits in maximum height, and which appears to be very suitable for a saddle-back form. Now the saddle-back construction itself is unfortunately too much ruined to allow its similarity to fig. 4 to be ascertained in more than a few particulars. It consists of a core of rubble and tafl, originally covered with a casing of limestone blocks; of the casing of the curved part only a stump, less than 35 cm. high, exists on either side. Most of the core still exists. Fig.5, for which I am indebted to Firth, gives a section of the building as seen from the east end; the solid black at each end represents the limestone casing.
The curve given in fig.4 is assumed to be 10 cubits wide; taking the cubit as equal to 52.5 cm., we have 525 cm. for the width. The total width of the saddle-back is exactly ascertainable; the distance between the vertical dotted lines in fig.5 is at the east end 557 cm., at the west end 556 cm., giving a mean of 556.5 cm. Here, then, we have a discrepancy of about 31 cm.
The ends of the curve where they rise from the zero line form the next most obvious points of comparison. In fig.6, the curve XZ shows the curvature, measured by Firth and myself, of the stumps of the saddle-back casing (at the east end, which is the best preserved part), and YZ is the curve established in fig. 3, from the shortest offset to the zero point. (See note below.) Here again there is a not inconsiderable difference.
The maximum height of the curve on the ostrakon is given as 3 1/2 cubits, say 52.5 cm. X 3.5 = 183.75 cm. Firth's section of the saddle-back, fig.5 , includes a conjectural restoration, based on the existing core, of the curve taken by the lost casing; this restoration is of course quite independent of any data afforded by the ostrakon. The maximum height of Firth's restored curve is equal to 174 cm. The correspondence is here very close, and the difference of about 10 cm. could easily be accounted for, if necessary, by slight denudation of the top of the core and by the fact that the thickness of the casing at this point can only be guessed.
For the rest, it will be seen by simple inspection; that there is a considerable resemblance between the general forms of the curve given in fig.4 and of the saddle-back curve as restored by Firth in fig.5.
In spite of the differences which have been pointed out, the degree to which the ostrakon and the saddle-back correspond, when added to the fact that the one was found in the closest proximity to the other, seem to leave little doubt possible that the diagram on the ostrakon was intended to serve as a guide to the masons in constructing the saddle-back. If this view be accepted, the discrepancies between the two will furnish interesting evidence of the amount of accuracy to which Zoser's builders worked, at all events when engaged on buildings of minor importance.
The ostrakon is now in the Cairo Museum, and is numbered 50036 in the Journal d'entree.
* Naturally another person, joining up the series of fixed points by a curved line, might produce a contour differing slightly from my own; but it is not possible to draw, from the data given, a curve which shall not be undulated and which shall fall on XZ.
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