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Conceived and designed the experiments: EZ IF IB. Performed the experiments: EZ YG IF IB. Analyzed the data: EZ IB. Contributed reagents/materials/analysis tools: EZ YG IF IB. Wrote the paper: EZ YG IF IB.

The authors have declared that no competing interests exist.

The modern metric system defines units of volume based on the cube. We propose that the ancient Egyptian system of measuring capacity employed a similar concept, but used the sphere instead. When considered in ancient Egyptian units, the volume of a sphere, whose circumference is one royal cubit, equals half a

Knowledge of the connection between linear dimensions and volume of containers is important, for instance, in order to achieve quick estimates of trade commodities. However, in many measuring systems, both ancient and modern, the length and volume units seem to have emerged independently, without a simple, intrinsic relation between them ^{3} make 1 liter

The use of the cube of a length-unit edge can be traced in antiquity in ancient Egypt. The Egyptian unit of length and volume were the royal cubit and

The cube of one cubit edge was used in ancient Egypt for estimating soil volumes in earthworks, see ^{3}/30 = 1

This cube-based relation was of little use in the typically ovoid-shaped Egyptian ceramic jars

According to dip test (see

Can the knowledge that the circumference of an ovoid-shaped container is 1 cubit assist in estimating its volume? Below we present evidence that the inherent relationship between ancient Egyptian units of length and volume measurements can be based on another elementary body – the sphere – and test our hypothesis based on the available archaeological information. We also demonstrate that the revealed relation was not relevant in Mesopotamia, where a different system of measuring length and volume units was in use.

In Egyptian units of length and volume, the volume of a spherical container of 1 cubit circumference would be 0.5

Substituting 1 royal cubit for c and employing the above-mentioned solution to Problems 41 and 44 in the Rhind Papyrus, one obtains

We checked the 1 royal cubit circumference

Despite variation in size, the most frequent maximal external circumference of these vessels (measured by us according to their drawings) indeed varies between 27 and 31 fingers (i.e., slightly above 1 royal cubit) and their modal volume, accounting for a wall width of 0.5–1.5 cm, varies between 0.45–0.65

According to the dip test both distributions are unimodal: p = 0.70 for the circumference and p = 0.43 for the volume.

Similar modal circumference and volume values were revealed by Barta

Beer jars were produced in the coiling technique

Globular pottery vessels – the best to demonstrate the 1 royal cubit circumference→½ ^{th} century BCE

We examined 89 Iron Age I-IIA Phoenicia-made globular jugs. Three of them we measured manually: one jug from Megiddo in the Jezreel Valley (

In this case, too, the distribution of the jugs' external maximal circumference has a clear mode at 25–30 fingers (

According to the dip test both distributions are unimodal: p = 0.83 for the circumference and p = 0.72 for the volume.

It is possible that the Phoenician globular jugs were used in trade of valuable liquids

In order to establish whether the sphere-based relation of 1 royal cubit circumference

According to the dip test the hypothesis that the distributionis unimodal can be rejected at p = 0.06.

One could have expected that the use of such formulae would have started in the Late Bronze Age, when the Levant, including Phoenicia, fell under direct Egyptian sway

According to the dip test the hypothesis that the distribution of the circumference is unimodal can be rejected at p = 0.11.

The ancient Egyptian 1 royal cubit

To conclude, the ancient Egyptian 1 royal cubit^{3}

The external circumference of a jar was estimated by direct measurement or by multiplying the length of the widest horizontal cross-section of a drawing by π.

In order to estimate the volume of a jar we scan the drawing, digitize its external and internal contours, and construct a 3D model by rotating its internal and external contours with Rhinoceros™ software. We can then estimate the volume of the jar, up to the neck, according to the internal contour. We estimate the wall width according to the drawings as well as by manually measuring the volume of three of the jugs and compared the result to the estimates obtained according to the digitized external profile. As we have demonstrated elsewhere

The unimodality of the distribution was tested according to the dip test