THE DIPOD

FROM THE MASTS OF THE SAILING SHIPS OF THE ANCIENT EMPIRE
TO THE TOOL TO LIFT AND HANDLE STONE BLOCKS

Bulletin de la Société Française d'Egyptologie, No. 140, Octobre 1997

Dr François CORRARD (1)(4)

Note of the translator:
This site is about the dipod, a two legged mast in Egypt Ancient Empire sailboats also a lift tool which may have been used in buildind pyramids.
This site reproduces a paper of Doctor François Corrard in the Bulletin de la Société Française d'Egyptologie no. 140, Octobre 1997. The original paper is also presented in French on this site and is the base reference with pictures and bibliographic notes. This is just an attempt to translate the French paper. The translator requests your understanding.
To access the French version in a separate window, kindly press the shift key and, without releasing the shift key, click on the French flag top right.

At the beginning of the Old Empire, Egypt, already the sailing birthplace in the Mediterranean (2), adopts for its sailing ships the dipod (3), this particular mast, which will make it possible to multiply by ten the functions of river and maritime transportation. In same time, the advent of the stone, which supplants brick, and its control will generate major buildings of the history of this country. In these two fields, mankind pushed its technical options until their limits. What were the interactions between them? It is known already that shipping was essential for the provisioning of the large building sites. How about the dipod? Has it been landed and used on earth?

I - THE DIPOD, MAST OF SAILING SHIP

1 - Its characteristics

Fig. 1. Boat V^th dynasty - Tomb of Ty

Formed of two wood trunks assembled at the top, the two feet drawn aside in a triangle, it is the highest mast of its time (5), (11 meters according to figure 1, in proportion of the characters). It is movable by a rotation around an axis joining its two feet, either lying on the boat, or set up. The feet are then attached to the same cross beam, sometimes resting against wooden squares, by large twisted and tended bonds. It forms thus, a particularly robust structure. Contemporary of the Old Empire, it appears with the III ^th Dynastie and disappears with the VI ^th that is a 500 years life (from 2700 years to 2200 years B.C.). Its development was such on the Nile, given the abundance of drawn testimonies, that it is practically specific of this Egyptian epoch (only some examples scattered of dipods are found elsewhere in the history of sailing,).

2 - This mast corresponds to a particular hull

Mast and hull are interdependent. To assemble a hull, there are two options:

The most common is to build a robust frame and to cover it with a thin coating: skin tended on a wood frame, bordered and adjusted on couples.
The Egyptians, singularly, chose the reverse, privileging a strong peripheral structure, broad and massive edges, to the detriment of the internal organization, very simplified, summarizing themselves with some bonds, slats and cross pieces.

This choice relies on the characteristics of the vegetable environment and of a faithful knowhow transmission.

Indeed, the first boats consist of papyrus bales, laid out at the periphery, in superimposed rows and firmly bound one with the others at regular intervals and at the two ends (6) (Fig.2).

Fig. 2. Boat made with reed bales, used on lake Titicaca (Museum of Ethnology Cambridge), probably very close to the ones used on the Nile

When the hulls are built with wood, the tree used is the Niloticum Acacia, described by Herodote (7), still of use at the beginning of this century (8). It is dense, will not putrefy, is filled with knots thus breakable and impossible to curve. It is cut off in thick boards, coarsely rectilinear of short length which are assembled one with the others like “bricks of one wall” using bonds and pegs, then worked to the adze (Fig.3). This assembly, like the boat of reed, has no or little internal frame.

Fig. 3. Assembly of the wooden edges as a brick wall - XII^th dynasty (L., D., II, 126).

The large funerary boat of Kheops (40 m long), made of cedar wood however easier to work, are assembled in the same way, each piece of wood with its immediate neighbors. The few cross pieces at the bottom of the hull do not go up until the top of the edges and do not have a relationship with the beams (Fig.4 and 5).

Fig. 4. Inside of the Kheops hull, being reconstructed. (photo G. Goyon)

The solidity of the hull is ensured by the only thickness of the edges (13 to 14 cm) on which these beams lay (9). Boats of Dahchour (10 m long), built according to same principles in the Middle Empire, have also thick edges (7 to 9 cm). These hulls, thus built without central skeleton, are flexible and deformable. To rigidify them, the Egyptians use the longitudinal tension of a large cable fixed at the two ends of the boat. The edges, while deviating, tighten the bonds (pegs and ropes) which maintain the assembly of each board.
This system is used

Fig. 5. Structure of the Kheops hull - IV^th dynasty. (B. Landström, Ships of the Pharaohs).

at the end of the construction (Fig.6), before fixing the transverse beams which will maintain the edges in their final form.
at sea (Fig.7), by giving some turns to the wooden axis, driven in the strands of the cable, to increase the resistance to the wave pressure.
on the Nile (Fig.1), when the wind becomes powerful, the sail pulls more at the top of the mast which, via the cables which maintain it behind (bastaques), raises the back of the boat (poop). Thus more the wind blows, more the hull rigidifies itself!

Fig. 6. Curving the edges before setting the beams - XII^th dynasty. (L., D., II, 126)

Fig. 7. Seagoing boat - V^th dynasty. (Sahuré, II, pl. XIII)

Some authors have thought that this longitudinal tension tended to curve the hull, or even to fold it, or even more to break it by the bottom, consequently justifying the need for the presence of a keel and couples. In fact, this longitudinal traction respects the bottom of the hull and curve the edges (Fig.8). Moreover, this curve of the walls of the boat is opposed to the pressure external of the water which tends to crush the hull. This deformation fulfills exactly the function of the couples. This technique compensates for the the internal structure weakness.

Fig. 8. Change in shape caused by the lengthwise cable tension.

In this flexible environment, the dipod has two assets:

its facility of operation, more stable around an axis than a point (case of a single mast), especially in the event of rolling on this narrow hull without a keel.
its triangular structure which has the mechanical property of being indeformable. This mast represents the most rigid part of the boat.

3 - To sail with a dipod

Before embarking, here are some simple navigation principles:

A sailing ship (Fig.9), moves at the same time in two fluids, water and air.
In water, the immersed part is subject to constraints which one can summarize as a force applied in a point, the drift center. It is located in the middle of the symmetrical hull.
In the air, we can, in the same way, add all the forces which act on the sail, resulting in only one force applied in a point, the sail center. It is approximately in the middle of this large vertical trapezoid.

The sailing ship, on the Nile, is tributary at the same time of the orientation of the wind and of the layout of the river. If they correspond, the mast is pulled up and the sails are hoisted. They are laid down in the event of head winds in order not to oppose resistance to oars and to the current which then remain the only actors of navigation.

Fig. 9. Force distribution. Mystic boat V^th dynasty. (Deshasheh, pl. VI)

Let us embark now for a voyage towards the south. Some blows of oars take us away from the bank. The sailors, at the back of the boat, orient with their oars the boat in the axis of the river. The north wind is present. We will be able to use the sail, and, first of all, to set up the dipod. This delicate operation, is usually accomplished in two steps (Fig.10) according to this only graphic testimony; to raise the dipod then to hoist the heavy sail. Another simultaneous method mast-sail could use the force of the wind (Fig.11). Indeed, the dipod is set up only when there is wind blowing forward from the back of the boat, therefore in the direction of gyration of the mast. Here is practically the operation; at the beginning, the sail is in place on the mast lying. The men by raising the end allow the wind to engulf itself in the sail and to finish the rotation. At the end, it is necessary to slacken a seil to soften the jolt then to tight it again to sail. Once the sail is set up, the back wind gives a very pleasant pace. The boat sails effortless (Fig.12). Indeed, without a keel, the least yaw spontaneously brings it back in the wind axis. There is no side force which may make it to lean or even to capsize. All is easy when the axis of the river is superimposed on that of the wind, which is frequent. When a light variation occurs, a correction by the oars rudders makes navigation still possible. But over 15°, it is necessary to bring down the sail and to use the oars. It is there, the limit of this navigation: 30° out of the 360° possible.

Fig. 10. Setting up the dipod - IV dynasty. (Tomb of Abibi at Saqqarah, Cairo Museum)

Fig. 11. Hypothesis to set up simultaneously the mast and the sail.

Fig. 12. Automatic adjustement to a small change of the back wind.

4 - Disappearance of the dipod to the profit of the single mast

Major changes appear (Fig.13):

The mast moves back until occupying the medium of the hull
The sail decreases in height, lengthens in width. The vertical trapezoid becomes an horizontal rectangle. The lower yard, which already existed with the dipod but lays down on the bridge, becomes suspended and stiffens the sail. Its leading edge is rectilinear. The sail center is at the vertical of the drift center. Sail and hull can swivel easily around this same axis.

How this change occurred? Is this the result of external contributions, conquests on ethnic groups of the South which would have used such riggings? Admittedly, the first representations of a single mast, before the Old Empire, come from Middle Egypt (Fig.14), Sudan (Fig.15) and Nubia but the clear offsetting of these masts of small size testifies of an exclusive use of back wind and thus does not constitute an innovation. In addition, conquests should have had brought a sudden and total appropriation of all the improvements. However, quite to the contrary, progressive and disparate modifications testify of a gradual research, of a constant inventiveness to be put to the credit of the Egyptians:

Fig. 13. Force distribution - New Empire. (B. Landstöm, Ships of the Pharaohs)

Fig. 14. Gerzean boat, about 3200 B.C. (vase of the British Museum n° 36326) fig 15

Fig. 15. Rock engraving - Sudan, 10 km south of the Egyptian border. (G.J. Verwers, 1962)

From the IV ^th to the V ^th dynasty, the mast moves back towards the center of the boat.
during V ^th dynasty, a single mast supports a trapezoidal sail (Fig.16).
during the VI ^th dynasty, single masts coexist with a tripod and this one carries a rectangular sail with suspended lower yard (Fig.17).

Fig. 16. V^th dynasty (B. Landstöm, Ships of the Pharaohs)
fig 17

Fig. 17. VI^th dynasty (Deir El Gebrawi, II, pl. VII, lower register)

Why this transition? Which is the stake?
With this modified rigging, the sail functions differently (Fig.18). It is not perpendicular any more to the axis of the wind but only tilted compared to this one. The airflows which are presented at the edge of attack will pass on both sides from the sail. Those which run out behind will accelerate because the distance to be traversed is a little longer than for those which run in front. Space between the molecules increases, the air is less dense generating a depression, whereas in front of the sail, the force of the wind generates a pressure. (18)
The resultant of all these pressures applied to the sail creates, (Fig.19)

first, a drift force perpendicular to the boat to which the immersed part of the hull is opposed,
and in addition, a force directed forwards which will make it possible the boat to sail perpendicular to the wind, and even to go up a little reverse to it. We arrive at this revolutionary concept, the boat pushed by the wind can from now on move slightly towards him.

This wind control increases considerably the driving freedom. The mast does not need more to be dismounted, to be so high and so robust. The dipod doesn’t have any more raison to be present. It has represented the quintessence of the transposition on board of the observation of the dead leaf carried by the wind. The human genius, in this case Egyptian, has intuitively transcended this natural fact. The disappearance of the dipod opens the way of the modern sail, that which we practise today.

Fig. 18. Forces on the sail.

Fig. 19. Forces on the hull.

II - THE DIPOD AS A TOOL TO LIFT AND HANDLE STONE BLOCKS

1 - Transport of the stone blocks on ground and on the Nile

Shipping transformed the Nile quiet force into an economic, political but also architectural power. Expeditions were numerous to the mines and careers of the South to supply the building sites of Pharaoh. These are the same men who transport the stone blocks from the extraction area to the boat, then who transport them on the river as sailors. The study of the dignitaries titles reveals that some of them associate nautical responsibilities - Chancellor of God, Lieutenant, Ordering crews, Leader pilot or Chief of the crews of the two boats - with the title of Director for great work for the king, the person in charge of a building site of Pharaoh. These two statutes can be simultaneous for some or successive for others. Ship management and building are thus much related. Couldn't a technique used on board have been used on ground? Could the dipod, with its so well controlled rocking movement, have been used on ground for lifting and handling of the stone blocks?

2 - The function of the dipod is identical, as a mast or as a lift tool

In the two situations (Fig.20 and 21), the system has two trunks of wood assembled in a triangle and the same rotational movement. On board, the tensions of bastaques behind, balance the force applied by the wind to the sail. On ground, the traction on the rope is opposed to the weight of the block. In both cases, the forces are exerted at its top and act in a perpendicular plan to the rotation axis.


Fig. 20. Dipod aboard.	Fig. 21. Landed dipod.

3 - How to use the dipod

The system provides a lift function the importance of which varies with the initial slope of the dipod compared to the ground.
Sustentation function

At start, the angle is of 70°. The block is maintained in sustentation close to the ground. A retaining cord opposed to the traction cord makes it possible to keep this position of balance. The block, guided by hand, can be moved for a few tens of centimeters, positioned with a high degree of precision, this with great facility.

Lift function elevation

The initial angle is 50 to 55°. Lifting is more important (approximately 30 cm) for example, to load or unload a block on a sledge. The required effort is more important. The addition of a second dipod decreases the tractive effort.
In both cases, the muscular effort is short, maximum at the beginning. More the dipod turns, more the tractive effort decreases and is null after a 30° to 40° rotation. The displacement of the men is short, of 1 meter to 1.50 meter.
Lastly, the dipod can:

put vertically a block,
and haul it on the ground by relieving it partially.

Use of two dipods
the addition of a second dipod decreases the effort required by at least one third, if it is (see Trigonometrical Appendix)

perpendicular to the cord traction
of the same length (isometric) as the first
even more, if it is higher than the first one (anisometric), so that the cord which joins them is perpendicular to the first, but the size of the wood trunks and their flexibility are limiting factors. It increases also the adherence of the system by being opposed to a slip of the legs of the first dipod.

4 - Experiments

We used as material:

a block of squarred limestone of a ton and half (1m 60 X 0.60 m X 0.70 m. density: 2.2)
rafters of fir tree 3.50 m long (section: 7.5 cm X 6 cm, weight 7.5 kg each one).
- Each dipod has two pieces of wood, tighted to one end by a simple cord, put in tension by the spacing of the feet.
a hemp braided cord with a 22 mm diameter, oversized for its use - its resistance is 3 tons - but preferred because of a better catch in hand and of its weak elongation. It girdles the stone block - with protected edges - and is suspended at the top of the dipod, at the intersection of the two pieces of wood with one turn on each piece to avoid slipping.

With only one dipod, we have, with much ease, moved, positioned (Fig.22) and set up the limestone block (Fig.23).
With two dipods, the same block was deposited (Fig.24) on a 15 cm thick pallet - supposed to represent a sledge - and unloaded (Fig.25). Eleven men mobilized this mass of a 1.5 metric ton.

Fig. 22. Sustentation. Positioning precisely the block by hand. On the left, the retaining cord. The traction cord is on the right. (single dipod)

Fig. 23. Setting up the block vertically (single dipod).

Fig. 24. Loading the block on a 15 cm thick pallet. (double isometric dipod).

Fig. 25. Unloading the block from the pallet. (double isometric dipod)

Use of a dipod chain (Fig.26)
Instead of two, we can align several dipods, in particular on the side of a pyramid, the leg in the angle formed by two bases, one row out of two. The rope goes in the higher angle of each intersection. It is hauled by the men placed on the higher base. On the steps, other workmen, two by dipod, maintain them horizontal, at the beginning.
The cable traction causes the the lower dipod rotation with the load. The rope, approximately parallel to the stone side, during this movement, deviates then approaches the second dipod, rests on the intersection of its arms and move it at its turn. The two wood parts of the first dipod, which is no more in function, are separated to make it flat and let pass the load which goes up without discontinuity.

5 - Place of the dipod on a building site

What are its specificities? Its extremely simplicity, its lightness, its great handiness, its ease of use and its interesting yield. These assets confer it a possible place besides the already indexed handling tools which are levers, wooden logs, brick scaffoldings, ropes and sledges.
As regards its lift use on the side of a pyramid, Herodote mentioned during its voyage - approximately 4 centuries B.C. - the use of machines made of short pieces of wood, laid out of step in step and making it possible to hoist the stones. Admittedly, for very heavy loads, the dipod does not compete with the interest of the sledge hauled on silt coated slopes. But, for stones of more modest size, it keeps all its interest to contribute to the construction.

Fig. 26. Lifting a block with a chain of aligned dipods. Model of the first steps of the Kheops pyramid at the same scale as the characters (54 0/00).

6 - Assumption or reality?

We do not have any element, this day, validating the use of the dipod on ground. Can one find traces of use, print of the two feet on the ground or notches of the rope on the edges of the blocks? Those of the colossus of Djéhoutihétep are protected during its transport. We can think that the stones of quality profited from this attention. As regards the more ordinary blocks, their unsmoothed faces and thus their nonrectilinear edges do not facilitate finding such traces.
Perhaps a day, we will have confirmation of the place of the dipod in the large building sites. Its assets confers it a place today from a practical and theoretical point of view.

TRIGONOMETRIC APPENDIX

We shall evaluate the traction force necessary to lift the block in three dipod configurations.
At each moment, the traction force and the weight can be decomposed in forces which are:

either directed according to the dipod axis, compressing the two pieces of wood,
or perpendicular to it, making it to rotate in one direction or another according to two opposite moments.

1 - Use of a single dipod

Calculations are made in the dipod mediator plane, that is the plane perpendicular to the dipode which contains its symetry axis.
We project in the mediator plane the two forces applied to the dipod - the weight and the traction force -.
We note respectively P et F the projected vectors norms.
The two angles a and b défined below are measured in the dipod mediator plane.
a is the angle between the dipod symetry axis and a horizontal line.
b is the complement (to 90°) of the angle between the projection in the mediator plane of the traction force and the dipod symetry axis.
h is the dipod height.
The moment vector caused by the weight relatively to the dipod rotation axis has a norm of:
Phcosa The moment vector caused by the traction force relatively to the dipod rotation axis has a norm of:
Fhcosb The instantaneous equilibrium is obtained by equaling the norms of these two moment vectors, that is:
Phcosa = Fhcosb From this, we deduce:
F = P cosa/cosb
This formula can be extended when two dipods of same height are placed in series with P replaced by a force.

2 - Use of two dipods, called ( I ) et ( I I )

At the equilibrium:
Pcosa = P₁cosb
F₁cosd = Fcosc
F₁ = P₁
From this, we deduce:
F = P x (cosa/cosb) x (cosd/cosc)

Two positions of the second dipod ( I I ) improve the yield - that is decrease F for a same weight P -

the second dipod is perpendicular to F, therefore cosc = 1
it has two possible lengths:

a) ( I I ) as the same length as ( I ). It's a double isometric dipod
therefore b = c
F = P x cosa
b) (I I ) is longer than ( I ), and of such length that the portion of cord AC be perpendicular to ( I ). It's a double anisometric dipod
therefore cosb = 1
F = P x cosa x cosd

3 - When the system rotates,

The angle a increases, F decreases in the three configurations:

single dipod: b decreases, therefore cosa/cosb decreases
double dipod:

isometric ( I I ) = ( I ), cosa decreases
anisometric ( I I ) > ( I ), d is constant and cosa x cosd decreases
thus

The maximum traction is at the beginning

More the system rotates, the easier it is to rotate

When the carrying dipod is vertical, the traction force is null

4 - Theoretical study of the effort to provide,

(depending on the weight P, the angle a and of the use of one or two dipods).

For an angle a initially small (50° - 55°), therefore to lift, it is worth to use if possible a double dipod, with unequal length (F initial = 20 to 40% of the weight).

For an angle a initially larger (70°), therefore to hold, move and position, a single dipod may suffice for a similar initial effort (F initial = 35 to 40% of the weight)

These angles are theoretical. They correspond to a position at which the dipod becomes really functionnal. In practice, the angle at start must be smaller than the optimal angle a to take into account the lengthening of the cord and making turns at the top of each piece of wood.

Notes

(1) My sincere thanks to Professor Luc PFIRSCH, to Professor Jean LECLANT, to Professor Jean VERCOUTTER, to my family and to all my friends who participated to the experiments.
(2) The oldest know representation of a rigging - mast, sail - is Egyptian, on a little Gerzean boat, at the end of the predynastic epoch, around 3200 B.C. The sail is a small woven mat, probably made with reeds. Vase of the British Museum no. 36326. See Fig. 14.
(3) This special mast that we call dipod is diversely named: goat mast and double mast by Boreux, two legged mast by Reisner, bipod par Landstrom. Etymologically, dipod looks to me more coherent with its two greek roots, derived from the word tripod of same origin, a nautical term used to designate a three legged mast.
(4) You are invited to send your comments to the author at the following email address:
fcorrard a t yahoo dot fr
You would, of course, change the above to something like: author@site.fr
(5) Assman, Das Grabdenkmal des Konigs Sa'hu-ré, II, Die Schiffsbilder, Leipzig, 1913, p. 149.
(6) Boreux Ch., Etudes de Nautique Egyptienne, Le Caire, 1924, p. 15, p. 182-184.
(7) Herodote, L'enquête Livre I à IV, Folio, Paris, 1990, Livre II, paragraphe 96.
(8) Clarke S., Nile boats and other matters, Ancient Egypt, Part I, 1920, p. 46-49, describes the construction of a naggr in 1911. Hornell J., The frameless boats of the middle Nile, Part I, The mariner's mirror 25, 1939, p. 418. Boreux Ch., Etudes de Nautique Egyptienne, Le Caire, 1924, p. 237.
(9) As a comparaison, a pomp boat, therefore for a similar use as the one of Kheops, built for Napoleon I - 22m long, kept in the marine Museum in Paris - with couples fixed every 40 cm, has broadsides of only 1.5 cm thick!
(10) Boreux Ch., Etudes de Nautique Egyptienne, Le Caire, 1924, p. 286-298.
(11) The word which means to travel use several signs, in particular, a sailboat with a spread sail to mean that the travel is southwards and a sailboat with a sail laid down to precise that the travel is northwards. Gardiner Egyptian grammar P. 585-586.
(12) The precision of the details of the Sahure boats (Fig. 7) permit several hypotheses regarding the way to erect the mast. Landstrom (Ships of the Pharaohs, New York, 1970, p. 66) proposes, same as Solver C.V. (Egyptian sea-going ships about 2600 B.C., The mariner's mirror, 47, 1961, p. 27-28) that the dipod rotates over a bench, helped by two large stone anchors resting at the extremities of its feet. Assmann (Das Grabdenkmal des Konigs Sa'hu-ré, II, Die Schiffsbilder, Leipzig, 1913, p. 146) proposes that a prop in a collar at 2/3 of the mast height goes through a small fork at the head of the boat before being towed. Borchard, as cited by Boreux (Etudes de Nautique Egyptienne, Le Caire, 1924, p. 486), proposes the idea of an horizontal cabestan, a kind of winch on wich is winded up the cord which pulls the mast.
(13) It seems that it is about erecting the mast and that two sailors, hands on knees, begin the rotation by an alternate movement, one holding the mast while the other goes kneeling down closer and lift it at his turn. The third sailor, who should go more to the front, finishes the rotation. This is difficult because of the little space left at the front of these boats.
(14) This principle is presently used on windsurfers. At start, the person is in the water, mast and sail slightly horizontal, waiting for the wind which in the same rotation, will set up the rigging and take the person out of the water.
(15) The rectangular sail and the suspended lower vergue appear for the first time in the Mereruka tomb, at the beginning of the VI^th dynasty. Landstrom (Ships of the Pharaohs, New York, 1970, p. 47).
(16) As proposed by Boreux (Etudes de Nautique Egyptienne, Le Caire, 1924, p. VI).
(17) In the Ancient Empire, the ratio between the distance of the mast to the front of the boat and the hull total length increases progressively from 28% to 41%. In the Middle Empire and in the New Empire, it is of approximately 50%.
(18) This phenomenon was analysed by the physicist Daniel Bernouilli only at the end of the eighteenth century.
(19) Bowen, as cited by D. Jones (Model boats from the tomb of Tut'ankhamun, Oxford, 1990, p. 55), estimates that sailboats of the XVIII^th could sail making an 80° angle with the wind direction, therefore in a 200° sector out of the 360° possible.
(20) See titulatures no. 5-16-43-49 as reported by Chevereau P-M. Contribution à la prosographie des cadres militaires de l'Ancien Empire et de la première Période Intermédiaire, RdE, 38, 1987, p. 15-20.
(21) The titles of Sebekhotep, at the end of the Ancient Empire, describe his career as a marine officer in charge of expéditions: Chancelor of God in the two large boats - Chief of the troops - Admiral - Captain of the Royal boat and at the end... Building site manager. Vallogia M., Stèle d'un chef d'expédition, BIFAO, 85, 1985, p. 264.

(22) Characteristics of hemp cord: (Corderie d'Or - Marseille)

      Diameter           Rupture load         Elongation 5%     
       (mm)              (metric ton)         with a traction of
                                              (metric ton)
        13                  1                    0,8
        22                  3                    2,4
        30                  6,6                  5,3
        44                 13,9                 11,1

(23) Herodote, L'enquête, Livre I à IV, Folio, Paris, 1990, Livre II, paragraphe 125.